acquiring the early a priori

Essay 6

Believers in the Land of Glory

Nature has the deep cunning which hides itself under the appearance ofopenness, so that simple people think they can see through her quite well,and all the while she is secretly preparing a refutation of their confidentprophecies. --- George Eliot1

(i)

At diverse moments within philosophy’s long career, grander views of itsprospects appear, which momentarily excite great enthusiasms yet eventuallydissipate under the cold scrutiny of sober fact. I believe that academic philosophy ispresently engaged in one of these cycles of great expectation, ambitions foundedupon the same mischaracterizations of descriptive practice as concern us elsewherein these essays. This new philosophical moment calls itself analytic metaphysics andpatterns itself after the proposals of the late David Lewis. Behind its projects lies aquasi-Kantian expectation that various metaphysical categories must exist that serveas fundamental prerequisites of descriptive thought. And we can distinguish twoseparate strands that emerge in support of these enterprises, based upon distinctflavors of semantic rigidity. The first of these is genetic in consideration andemphasizes the conceptual prerequisites we allegedly need for making sense of theworld about us. For example, early in our linguistic training, we learn to reasonabout the “parts” and “wholes” of solids and liquids in innocuous manners that rarelyprovoke controversy within the rounds of everyday affairs. Likewise, we acquirejuvenile facility in discussing how “causes” should relate to“effects.” Relying upon these basic categories, we canunderstand what it means when we are informed, at a laterstage of education, that recondite “very small parts” ofmatter (viz., atoms and molecules) cause the the high leveleffects we witness in ordinary life. Let’s dub the reasoningcapacities so acquired the early apriori. FIG:ACQUIRING THE EARLY APRIORI Such inferentialskills encourage the supposition that an armchair disciplineof metaphysics might isolate the essential tenets concerning“parts,” “wholes” and “cause” central within these capacities. L.A. Paul praises the

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benefits to be obtained from this dedicated study as follows:An ontological theory of parts and wholes (a mereology) of physical objectsdescribes more fundamental and more general constructional principles thanphysics or chemistry does, for it gives general principles that govern all thephysical objects with parts, including microparticles, atoms and molecules. For example, chemistry may tell us that the physical structure of apolycarbonate is causally created by arranging elements a certain way, andthat its physical parts consist of these arrangements of elements and theattractive forces between them. Mereology contributes the additional claimthat the molecule just is (say), the mereological fusion of its arranged parts(the elements and the attractive forces). The polycarbonate molecule iscreated by this mereological fusion, but not in a causal sense. Rather, it iscreated in the compositional or ontological sense: it exists when the partsarranged in the right way exist. So the metaphysics tells us what it is to be asum or physical object composed of these structured arrangements of parts,and thus tells us how the physical object is metaphysically constructed(composed) from its parts. In contrast, chemistry tells us what some of theparts and the arrangements of the parts are for different kinds of molecules,and it also tells us how to causally manipulate the world in order to bringsuch arrangements into existence. Chemistry tells us how molecules arephysically or causally constructed from attractive forces and smallerparticles.2

Here the term “mereology” signifies the branch of apriori doctrine allegedlydemanded by our everyday talk of parts and wholes. In this essay, we shall belargely concerned with the parallel necessitarian tenets offered on behalf on causeand effect, but allied considerations with respect to mereology have been brisklycanvassed in Essay 1.

But there is a second pillar of support behind the analytic metaphysics projectsthat leans more heavily upon forward-looking considerations of scientificdevelopment, rather than to the prerequisites of early learning that Paul emphasizes. Consider, in this light, the following extract from Ted Sider wherein the tenets ofmetaphysics represent a conceptual prescience upon which any properly developedscience must build:

A realistic picture of science leaves room for a metaphysics tempered byhumility. (Just like scientists, metaphysicians begin with observations, albeitquite mundane ones: there are objects, these objects have properties, theylast over time, and so on. And just like scientists, metaphysicians go on to

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construct general theories based upon those observations, even though theobservations do not logically settle which theory is correct. In doing so,metaphysicians use standards for choosing theories that are like thestandards used by scientists (simplicity, comprehensiveness, elegance, and soon). Emphasizing continuity with science helps dispel radical skepticismabout metaphysics; the humility comes in when we remember thediscontinuities. Observation bears on metaphysics in a very indirect way,and it is far less clear how to employ standards of theory choice (likesimplicity) in metaphysics than it is in science. But metaphysicians can, andshould, acknowledge this. Metaphysics is speculative, and rarely if everresults in certainty. Who would have thought otherwise?... The philosophermust therefore live with uncertainty whether her life’s work is ultimatelymeaningful--that is the cost of the breadth of reflection demanded byphilosophy. Philosophy’s reflective nature is generally a good thing, but thedown side is that it can lead to paralysis. Don’t let it. You don’t need tohave answers to all meta-questions before you can ask first-order questions.3

Of these two lines of thought, my own sympathies align more closely withPaul’s early apriori themes, for, as Essay 1 emphasizes, we require basicvocabularies for managing the many varieties of explanatory strategy we must exploitif we hope to capture worldly behaviors within our descriptive snares. To theseends, we sometimes must articulate novel reasoning architectures clearly to ourselvesand to others who can benefit from these routines. The phrases “cause” and “partof” contribute significantly to these policies of focused language arrangement. Butany further consolidation into a structured apriorist metaphysics becomes blocked bythe fact that the specialized architectures that we arrange through appeals to “cause”and “part” immediately appropriate these same words to localized descriptivepurposes, as we shall observe with respect to “cause” soon. The end result is that,over time, our managerial phrases develop evolving and contextually sensitivesemantic attachments that resist clean analysis of the necessitarian cast that Paulexpects within a metaphysics.

(ii)

The easiest way to explain the processes I have in mind is simply to observehow readily “cause” adjusts its semantic bearings when we move from one

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marching method computation for an evolutionary model

explanatory architecture to another. To this end, let us start with a paradigmaticemployment” of “cause” and study how readily it shifts emphasis as its surroundingdescriptive architecture is adjusted. To this end, we can employ one of this book’schief (over-)utilized examples, wave motion in a violin string. We model suchcircumstances with the equation 2y/t2 = k 2y/x2 with Dirichlet boundaryconditions y(0,t) = 0 and y(1,t) = 0 for all t, indicating that our string stays fixed atbridge and nut. Suppose that at the initial time t0 we release the string in a simpleplucked pattern f(x). FIG: MARCHING METHOD COMPUTATION The waveequation describes the causal processes that carry out system forward in time, viz.,the constant kdetermines the rate atwhich the potentialenergy stored within thestring’s curvatureconverts to kineticacceleration and backagain as the string movesback and forth across itsaxis. The result is themost characteristic formof causal process: thewave motion that drivesour initial pulse forwardin time. An old canardstill repeated within philosophy of science circles maintains that physics has no needof the notion of “cause.” But this claim is simply mistaken and results from blurringtogether explanatory settings of markedly different characters. As Essay 2 explains,the differential equational models that directly capture genuine causal processesusually possess a formal feature (hyperbolic signature) absent within equational setsthat capture non-evolutionary physical circumstances (such as equilibrium condition,generally of elliptic signature). Viewed from this perspective, standard in modernbooks on partial differential equations, it seems odd to declare that “physics has noneed of ‘cause’” when modeling equations of an appropriate character directlycapture the evolutionary developments that we expect from a causal process.

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five point stencil

However, these issues are complicated by the fact that somewhat sophisticatedmathematical means are required to capture the root notion of “causal process”precisely, viz. hyperbolic partial differential equations and their more sophisticatedfunctional analysis cousins. But, obviously, we obtain a pretty good conception of theroot idea earlier in life--how do we manage this? Answer: we employ what amathematician calls a “finite difference” rendering of the causal connections underreview. This simply means that we characterize the infinitesimal relationshipsencoded within our differential equations in terms of finitary little “steps” of spaceand time (Δx, Δy, Δz and Δt). For example, 2y/t2 = k 2y/x2 claims that ourstring’s vertical acceleration elevation 2y/t2 can be calculated by multiplying itscurrent curvature 2y/x2 by k. At a given time t, we can approximate the currentcurvature at point x with a horizontal block of three spatial values in the neighborhoodof x (viz., the string’s elevations at x, x - Δx, and x + Δx, all considered at time t). Likewise x’s acceleration at t can be captured with a vertical block of three temporalvalues (viz. the string’s elevations at t - Δt, t, t + Δt, all at position x). We can then

estimate the string’s x position at time t+ Δt by insuring that the three verticalvalues match the three horizontal valuesmultiplied by k. This computational ruleis called a five point stencil and allows acomputer to track how the developingcausal processes in our string will unfoldon graph paper, in the mannerillustrated.4 FIG: FIVE POINTSTENCIL A scheme that progressively

fills in its little finite difference squares via a “time forward” algebraic rule such asthis is usually called a marching method (corresponding, I suppose, to the celebratedMarch of Time).

When we render these finite difference instructions into English, they turn intopropositions of exactly the “cause now, effect later” form beloved of Hume and hismany calculus-avoiding philosophical descendants, who presume (without muchattempt at verification, insofar as I can determine) that “laws of nature” invariablytake the form “for all x and all t, if F(x) holds at t, then G(x) will hold at t + Δt).” There is much to be regretted in loose assertions such as this, but I will reserve thebulk of my complaints to other essays. However, many philosophers of a “sciencehas no need of causal process” persuasion argue for this position by observing thatmodern science rarely expresses its “fundamental relationships” in Humean “cause

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calculating an arrow’s path by marching method routine

first; effect later” terms. Exactly so, but that’s simply because true causal processesare best captured in terms of differential equation relationships that we can expressonly in approximate terms if we don’t know the calculus.5

With respect to our childhood acquisition of early apriori skills, we likewiseregister nature’s causal processes to ourselves in finite difference terms at first,despite the fact that, from the vantage point of a more exacting conceptualdevelopment, this juvenile policy merely gestures at the proper core notion through asurrounding swarm of approximation space surrogates. Heedless parents rarelyinstruct their children in the calculus at a sufficiently tender age.

Incidently, situations in which important physical conceptions first appear inthe guise of approximation space surrogates are rather common in the development ofscience, for it often takes mathematics a long time to develop adequate resources forcapturing the central notions in fully satisfactory terms.6

For later use, let us note that an even simpler marching method technique canbe used to compute an arrow’s flight after it leaves the bow. FIG: CALCULATINGAN ARROW’S PATH Here anatural modeling utilizes anordinary differential equation(ODE) of evolutionary typerather than the partialdifferential equation (PDE)required for our string. The keydifference is that we don’trequire any data about ourarrow’s spatial neighbors to fillout our charts, but simply twobits of temporal data. Our 5-point stencil rule collapses into the simpler 3-point stencilroutine commonly called Euler’s method.7

(iii)

Although the notion of causal process, paradigmatically exemplified by wavemotion of any kind, qualifies as an important and estimable scientific notion, it hardlyexhausts the many descriptive utilities that the word “cause” executes on our behalf. And many of these supplementary skills form vital parts of our early a priori training(and doom it later to its inevitable decline from necessitarian grace). A key themerunning through all of these essays maintains that the behavioral complexities of

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Fourier change in descriptive basis

nature force us to forever devise clever strategies that allow us take advantage of ourcomputational opportunities when we sees ‘em.8 Such physics avoidance techniquesgenerate in their wake a bewildering array of explanatory architectures that we mustdistinguish and talk about when we to impart methodological wisdom to our peers. We likewise require basic management tools to keep track of strategic adjustmentsfor ourselves when we redirect old reasoning skills to new purposes. So let usinvestigate how our friend “cause” behaves when we engage in explanatory shifts ofthis kind.

We can do this simply by considering our violin string from a different point ofview, which often offers deeper insights into its behavior. Specifically, a marchingmethod routine only perform adequately as long as the energy inputted in the stringremains trapped in traveling wave front packets as illustrated in the graphs above. More commonly, however, a variety of interactions9 redistribute this energy acrossthe entire string in a manner that, after a short relaxation time, relocates the inputtedenergy within the lowest rungs of the string’s overtone spectrum. That is, the energybecomes allocated to the individual modes of sine wave vibration that we associatewith the string’s fundamental tone, the octave above, the fifth above that and so on. After this energetic resettlement occurs, the wave front individualities characteristic ofthe string’s original plucking, bowing or hammering become lost to view and anexcellent descriptive opportunity emerges in their stead, based upon the fact that wecan now characterize our string’s movements as a superposition of standing wavepatterns that retain their individual energies for significant periods of time.

This form of altered representation is calledFourier analysis and can be pictured as a shift ofbasis vectors within a common descriptive arena,shifting from the position representation (that workswell for traveling wave fronts) to an energyrepresentation (that works better for standing wavepatterns). FIG: FOURIER SHIFT INDESCRIPTIVE BASIS

I strongly emphasize that, in mostapplicational circumstances, the Fourier treatment

supplies far more reliable answers than an evolutionary approach employing marchingmethod reasonings, due to the latter’s delicate vulnerability to numerical error. Butthe explanatory landscape in which a Fourier treatment unfolds embodies a number ofsignificant contextual features absent from the context of a straightforwardevolutionary modeling.10

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eigenfunctions for unloaded and loaded strings

The basic trick is to exploit the conservation of the energy allotments assignedto each of the Fourier-basis descriptive variables. This focus allows us to pushsignificant temporal aspects of our problem out of view. But how can we know thatthis possible? Well, a proper answer is subtle11 but the upshot tells us that thebehaviors of a wide range of confined systems without significant dissipation can befactored into periodic sub-motions that continually return is their starting positionswith no intervening loss of energy. A simpler prototype for this cyclic pattern ofenergetic expression is the swinging pendulum. At its two turn-around points, thependulum’s energy is expressed entirely in the form of gravitational potential energy,which operates to move the bob back towards the central axis. As the bob crossesthis line, its energy becomes entirely kinetic in expression, causing it to overshoot itsnatural rest position. A similar energetic exchange occurs within each vibrationalcomponent of our violin string, except that the potential energy storage assumes theform of internal strain energy dependent upon the string’s curvature.12 Unlike thependulum’s single form of cyclic storage, our string contains a discrete spectrum ofstanding wave processes, all superimposed upon one another. As a result, the fullmovements of our string will appear very complicated, but the Fourier factoringshows that, in the absence of dissipation, it is actually cycling through a number ofsimple, simultaneous processes.

The configurations that a normal violin string mode assumes when it sits at oneof its potential energy turn-around points are familiar to all; they are simply the sinewaves of simple tonal analysis. But if we tinker slightly with our string model, byadding some uneven weights here and there,13 the altered system may store itspotential energy in altered patterns, such as the loaded irregular curvature illustrated,which is no longer a true sine wave. FIG: LOWEST EIGENFUNCTIONS FORUNLOADED STRINGS To figure outwhat these altered storage patterns arelike, scientists first address ourselves a“time removed” question14 akin toascertaining the positions at the bottomof the hill where Jack and Jill will cometo rest after they stop tumbling, viz.,

(A) What will our variousvibratory modes look like at their turnaround points when their storedenergies are expressed entirely in a “purely potential manner”?

In the jargon of the mathematicians, this preliminary quest for our system’s reservoirsof potential energy is called an eigenfunction problem. When we turn to questions

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shooting method for a control problem

such as this, the explanatory landscape in which our reasonings unfold shifts subtly.15 Well, we need a strategic trick to address this question profitably, so let us

return to our archer and his arrow and see if we can’t find a way of answering ourstring question by tweaking the marching method reasoning we applied to the arrow’sflight in a clever way. Right now we know how to answer query B:

(B) Where on the target will an arrow shot at position x and angle α withvelocity v be caused to land?

The following planning question lies in the general descriptive neighborhood of B butrequires a slight smattering of elemental strategy for its answer:

(C) What initial angle α will cause an arrow shot at position x with velocity vto land in the bull’s eye?

Plainly we can apply the marching method reasoning utilized for (B) to address thisnew problem by starting with a trial shooting angle θ0, correcting this initial accordingto the error obtained in hitting the target and repeating the marching method routineover a sequence of improving θi guesses until we find the proper θf is found. This sortof technique is usually called a shooting method for obvious reasons and computerengineers often build them by inserting off-the-shelf marching methods (e.g., a Runge-Kutta scheme) into a suitable feedback programming loop. FIG: SHOOTINGMETHOD FOR A CONTROL PROBLEM

Mathematicians classify(C) as a control problem (ratherthan an initial value problem likeB). With a little more tinkering,we realize that we can calculatethe equilibrium shape of a loadedclothesline in a very similarmanner, by asking, parallel to(C):

(D) What initial lefthandslope α will allow theclothesline to sag in exactly the manner required for storing the least amountof potential energy (calculated according to its curvature)?

Now our original task vis a vis our string was to answer the question: (E) Which maximal potential energy shapes will cause our string to preserveenergy when excited in that mode?

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calculating clothesline shape by shooting method

explaining a transferred reasoning method

If we now apply standard separation of variable techniques found in any differentialequation textbook, we realize that the sundry potential energy conserving patterns ofour loaded violin string must obey the eigenfunction equation 2(k(x)y(x))/x2 = -λy(x) for integral values of λ (the value λ = 1 corresponds to our lowest vibratorymode). But this equation is formally nearly identical to that for a loaded clothesline,so it’s plain that we can plot out the desired shape through shooting methodtechniques. FIG: CALCULATINGCLOTHESLINE SHAPE

Throughout these essays I lay heavy stresson the human need to reorient old reasoningschemes swiftly to novel purposes and thateveryday commonsense thinking contains manyof the strategic management tools required tocarry out these strategic borrowings. In thisspirit, observe that most of the strategicreorientations illustrated in our succession ofexamples strike us as entirely commonsensical policies to follow (except, possibly,the move to (E)). Indeed, mankind has long exploited such tactical adjustments toredirect old reasoning skills to fresh purposes and various primordial ancestors musthave recognized the utility of our archer/clothesline transfer. As indicated in Essay 1,implementing these reasoning shifts effectively requires basic capacities in languagemanagement, such as the phraseology our clever caveman would have needed toadvise his aboriginal audience that a knowledge of clotheslines can be usefullytransferred to bow and arrow circumstances.16 FIG: EXPLAINING ATRANSFERRED REASONING METHOD

Note that the word “cause” appears in all of our (A) to (E) problems, conveyedthere by the natural currents of conceptual linkage that tie together the underlying

reasoning methods. However, from areferential point of view (viz., how theword relates to physical reality), aremarkable shift has occurred, becausein (B- E), the word no longer attachesto any evident causal process in themanner of (A).17 In particular, in (B)our attention has shifted to features ofthe central “control variable” involved,our archer’s shooting angle. In fact,

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my colleague Jim Woodward, through an extensive survey of real life protocols, hasdemonstrated that a large amount of everyday causal talk, within both science andeveryday life, centers upon the outcome of various possible manipulations upon targetvariables (often one at a time). As we’ll shortly see, Woodward’s manipulationistconcerns play a central role in planning (and explaining to others) how a wide varietyof novel forms of explanatory architecture should be pieced together.18

Returning to our original motivations for describing our string in a Fourier-likeway (viz., as a superposition of energy conserving modes), we observe a prettyparadox. We were originally drawn to the technique because of its strong physicsavoidance virtues--the invariant nature of our string’s modes allows us to push mostissues of temporal development off the table and to instead concentrate upon the time-removed question of what these eigenfunction modes look like if frozen into place. Incontrast, the notion of causal process that we explicated through appeal to continuouswave progression inherently focuses upon temporal alteration, the wave motions thatcarry a system forward in time from one state to another. But the explanatorylandscape of our Fourier-based tactics avoids direct consideration of thesedevelopmental processes, by asking questions that are largely free of temporalingredients such as (C - E).19 But notice that the word “cause” reappears in both (C)and (E). In Essay 2, I analogize this lexical persistence to the fabled cat thatcontinually returns home despite all efforts to get rid of it (“The cat came back/Wethought he was a goner/But the cat came back/The very next day”). So why does“cause” act like that damnable cat?

The answer lies in the nature of our commonsensical or early apriori training,which was inherently jumbled to begin with. Indeed, basic questions of manipulationcontrol are palpably of greater salience to self-centered youngsters than autonomousevolutions, from which they must be weaned by patient appeals to reasoningadjustment (“Take yourself out of the picture and just ask where that arrow will go ifit’s aimed at a certain angle”).20 In other words, there’s no clear chicken and no clearegg within the developmental etiology of “cause.” As we gradually learn to make ourway through this vexing world of ours, we (and our caveman ancestors) must alsoacquire the rudiments of strategic borrowing early on, including an ability to followmost of the “commonsensical reasoning” adjustments just illustrated.

Portents of the early apriori’s eventual downfall can already be discernedwithin these innocuous juvenile allowances for reasoning transfer.

(iv)

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a proposal for maintaining the system statically

Manipulationist (or controlling variable) considerations often prove centralwithin everyday applications of “cause” because many basic forms of effectivemodeling strategy rely upon isolating special collections of “if ...then” claims of aWoodwardian character--I’ll call these “manipulationist conditionals.” So it is notsurprising that the word “cause” frequently serves as our chief term of choice inmanaging these arrangements. I’ll briefly outline two examples that are discussed ingreater detail in Essays 7 and 5, respectively.

Example I: A central technique of physics avoidance is the exploitation ofconstraints, which represent prior information about the eventual behavior of a targetsystem, usually resolved on a coarser observation scale than other aspects of onesmodeling effort. The great paradigm for data usage of this type is found within theappeals to rigidity within mechanics, formally registered as “side condition”requirements that the distances between various points remain invariant. Anothervital tactic of physics simplification, just reviewed in connection with our standingwaves, is that of removing temporal considerations from the table, at leasttemporarily. With respect tothe linkedseesawsillustrated,Lagrangedemonstrated thesignificantinferential advantages obtainable from first mapping out, in a time-avoiding manner,how the statics of our seesaws operate, that is, determining where and howcounterbalancing forces should be applied to hold the arrangements in balance. FIG:A PROPOSAL FOR MAINTAINING A SYSTEM STATICALLY Only afterward,should we consider its dynamic circumstances, that is, how the gizmo freely moves,building upon our preliminary statical knowledge. The questions we ask to thissecond purpose are time-free in character; only later do we lift these considerationsinto an evolutionary frame.21 But the statical knowledge needed requires that weknow, for every permissible configuration of the seesaws, what holding forces canmaintain the apparatus in a selected configuration selected. Prima facie, this is not aneasy question to answer. Lagrange’s principle of virtual work proposes that we begin

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convergence to a fixed point

with an arbitrary proposal for these balancing forces (Fi for each child) and thenwiggle various children one at a time, pulling the rest of the system along with theadjustment.22 If the manipulation δ on child i does not generate an exactly opposingcounter response from the internal arrangements of the seesaws, our trial set of staticholding forces was not correct (a physicist would say that “the virtual work of thevariation does not vanish”). If so, we should correct the trial force Fi to a bettervalue. We should then repeat the same test onanother child j, whose correction may force us torevisit our Fi correction again. But if we’re lucky,these virtual work variational tests will eventuallyconverge to a correct schedule of static holdingforces as a fixed point result. FIG:CONVERGENCE TO A FIXED POINT

What is the general nature of the data weassemble by approaching our linked teeter-tottersin Lagrange’s manner? They all involve evaluations of the form: “if child i is movedthrough a vertical distance δ in a virtual manner, this manipulation will cause Xamount of opposing work to arise as a reaction on behalf of the rest of the system.” As Essay 7 explains, a mathematician regards facts of this general character ascomprising a “space” with an an evaluative “norm” that can direct us, through animproving set of approximations, to the desired statical answers. But these guidingvirtual work facts can be immediately identified as a set of manipulation conditionalsof the sort Woodward has highlighted in his researches. From a reasoningarchitecture perspective, Lagrange’s variational techniques allow us to blend twodistinct classes of fact with one another: our partial rigid constraint data (the arms ofthe seesaws remain rigid) and considerations pertinent to the active forces on display(viz. gravitation and the connecting spring). But blending together these two fonts ofphysical information carries subtle data harmonization problems in its wake, whichLagrange sidesteps through his celebrated “multiplier” technique. In explicating therationale behind Lagrange’s reasoning policies, physicists speak of causation actingacross scales as justification23: “the rigidities binding the system together upon amacroscopic level cause their molecular counterparts to oppose the inactive forcesacting upon the system perfectly, producing a net cancellation of effective work thatallows us to sidestep any direct consideration of these factors in our computations.” As we’ll later see, a special space of Woodward-like manipulation conditionals playsan important role in these integrative operations.

Example II: A related association with “cause” with a different form of

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felsic volcanic rocks

reasoning architecture is encountered within thevarieties of multiscalar modeling discussed in Essays 1and 5. A fond memory of ones childhood rockcollection suggests typical examples. FIG: FELSICVOLCANIC ROCKS There are a wide variety ofrocks of essentially the same chemical composition asgranite (they all derive from an identical volcanicmagma) that possess quite different material propertieson a macroscopic scale--compare granite with rhyolite,pumice, obsidian or gneiss. Their behavioraldifferences (which are significant) trace to the mannerin which the lava was tossed about and cooled long ago and, in the case of gneiss, thehistory of how the solidified glob has been stressed and reheated subsequently.

Although formulating a single level mechanics commodious enough toaccommodate such complex materials represented a longtime goal of physicists suchas Pierre Duhem and Clifford Truesdell, recent advances in materials science andcomputing have found that constructing a multi-layered reasoning architecture thatmimics the natural scale separations found in nature often provides a more effectivereasoning policy.24 Here we assemble a collection of coordinated submodels thatdirectly reflect the fact that, e.g., the dislocations inside the grain within graniteinhabits regimes of characteristic scale length 1 μm, the mineral grain itself lives ataround .1 mm, and so on. Sample exemplars of such a multiscalar modeling aresupplied in Essays 1 and 5; they consist in collections of linked submodels thatcommunicate with one another through homogenization techniques rather thanstraightforward amalgamation of data. FIG: MULTISCALAR ARCHITECTURESFOR GRANITE AND PUMICE Relying upon those essays for details, the diagramillustrates how distinct reasoning architectures might be assembled for pumice andgranite, in a manner that directly apes the vital layers of higher scale structure thatdistinguish one rock from another. The basic trick is to assemble submodels thatcapture the dominant causal events that one witnesses within the material at eachcharacteristic size. For example, the different architectures sketched reflect the factthat the mineral grain in granite usually shears elastically under stress but sometimesrecrystallizes; that pumice possesses no significant grain but contains lots of trappedgas bubbles that sometimes burst when their obsidian walls melt under highertemperatures. And so forth. Crucial within these arrangements are the ways in whicheach component submodel sends information across scales to the other submodels;these generally invoke some manner of asymptotic averaging, rather than simply

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multiscalar architectures for pumice and granite

passing unprocessed conclusions forward from one scale to another. Engineerssometimes explicate these homogenizations in visual metaphors; they say that, whenthe dislocations pile up at a grain wall, the enclosing crystal “sees” these adjustmentsas an increase in brittleness. Equally commonly they talk of cross-scalar “causes”--the dislocation pileup causes the grain to become brittle. And such causes alsooperate in a downward direction--repeated macroscopic loading and unloading causesdislocation pileup. The basic architecture of a multiscalar modeling directly reflectsthese homogenization/causal dependency relationships. Accordingly, successfulhomogenization policies rest upon our abilities to compile a rich library of reliable“how one scale affects another” conditionals. In addition to these basic dominantbehavior facts, we need additional conditional data with respect to the exceptionalcases that sometimes disturb normal interscalar patterns. Example: a severedislocation pileup will cause the large scale rock to fracture, rather than simply

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becoming more brittle, as usually happens. This is the empirical data we utilize indevising the corrective feedback loops in our scheme, marked in the diagram by large,upward-pointing arrows.

In our commonsense thinking about such matters, we employ “cause” tostructure our thinking in closely related ways–if we heat up pumice a lot, some of itsobsidian walls may melt, causing some of the trapped gases to push outward, furthercausing an increase in macroscopic volume.25 But if confining pressures are too great,that pressure may cause the mobile gas to diffuse into the melted rock. And so forth. Once again observe that these invocations of “cause” operate across scales in bothbottom-up and top-down manners here, linking together the localized scenarios inwhose terms we usually conceptualize the behaviors of complex materials withineveryday life.

The pertinent lesson to be extracted from these examples is that we commonlycompile appropriate sets of Woodward-like conditionals26 before we are ready toarticulate plausible “laws” (i.e., adequate constitutive principles) for our variousrocks. In this fashion, “cause” and “effect” talk serves commonsensical purposes byarranging the components of an effective reasoning policy into a workable modelingarchitecture. If, like our caveman, we wish to communicate our newly discoveredmodeling technique to our chums, talking of interscalar “causes” is a excellent vehiclefor highlighting the key structural ingredients needed.

In these circumstances, the applicational signification of “cause” registers acomplicated mixture of structural and physically descriptive information, dependingupon the exact strategic context to which the word is applied. In other developments(see Appendix 2 of Essay 2 for an example), the physically descriptive componentscan sometimes wither away entirely, leaving only the management aspects active.

(v)

As already observed, we want to mimic genuine physical relationships inselecting our submodels and their interscalar messaging. Many of these reflect, incritical ways, fundamental issues of coherence and randomness. What is the causaleffect of stretching a section of iron spring with a pressure or tension rather thanheating it (both operations inject additional energy into its interior)? Answer: theinput pressure possesses a directional coherence that can be transferred to themolecular lattice of the iron through stretching the bonds between them when weelongate the spring. FIG: HOW A COHERENT PRESSURE CAN BE STORED AS

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how a coherent pressure can be stored as internal energy

INTERNAL ENERGY However,if we’re clumsy, we may notstretch the spring in acoherne direction at all but insteadtransfer all of our input energydirectly to the jigglings of theindividual molecules, in whichcase our energetic interventionwill have merely heated up thematerial and none of our exertionwill increase the spring’s strainenergy. Plainly, we can extract alot more useful work from astretched spring than a merelyheated one, due to the fact that the intermolecular cooperation captured within thestrain energy conception allows the spring to “remember” the coherent effort withinits original stretching, which it utterly “forgets” if it merely heats up.27 These are thedistinctions of which thermodynamic thinking is comprised and help explain why thesolid behaviors of the world tend to divide into fairly distinct layers of behavioraldominance (insofar as a manipulation heats up a material, it loses some of the crucialwork capacity it might otherwise enjoy at the relevant characteristic size scale).28 Soit is not surprising that Woodward-like issues of manipulations and controllingvariables are frequently central in mapping out how a novel reasoning architecturemight effectively mimic some of nature’s own dependency relationships.

Observe, however, that I’m not saying that “cause” in general means anythingsuch as “induces a focused change at a lower or higher size scale,” for the word withequal centrality attaches itself to autonomous causal processes of the sort typified bywave motion. Insofar as I can see, the seeds of all of these specializations can befound in the ragtag collection of management skills that we acquire during our earlyapriori training.29 The evolved result resembles Wittgenstein’s remarks on the generalconcept of “number”:

And we extend our concept of number as in spinning a thread we twist fiber onfiber. And the strength of the thread does not reside in the fact that some onefiber runs through its whole length, but in the overlapping of many fibers.30

But these same initial seeds signal the eventual demise of persistingmetaphysical doctrine, in the fashion that Paul hopes to isolate. Any rigidrequirement on how “causes” must relate to “effects” is apt to crumble as these words

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rebuilding science while at sea

naturally guide themselves into local architectures where distinct bonds betweenwords and world subsequently solidify, in accordance with the descriptivepracticalities facilitated by those local usages. As our other essays illustrate, such isthe eventual semantic fate of many other words as well. For our human purposes, it’swell that evolving usage silently proliferates in this way, for nature’s multifariousbehaviors presents us with descriptive challenges that require great methodologicalsuppleness in concocting reasoning schemes that can ably exploit her computationalopportunities wherever they emerge.

On this score Paul and her fellow analytic metaphysicians have been betrayedby the “rigidified semantics” thinking favored within mainstream endeavor withincontemporary philosophy of language, whose founding assumptions are criticized inEssay 1. According to these expectations, during our early apriori training we attacha central “meaning” to “cause,” with a firm extensions in all “possible worlds.” Westick with this basic frame of word/world(s) attachment thereafter, although we maydevelop satellite usages based upon this core usage. As Paul conceives themetaphysician’s task, she should frame a “theory” that detaches the original causalur-concept from any satellite moss that it may gather later.

W.V. Quine, of course, criticizes such apriorist and necessitarian assumptionson roughly the same grounds as I have, viz., the natural processes of linguisticdevelopment rarely leave any foundational stones unmoved. To these ends, hesummons his celebrated Neurath’s boat metaphor--the vessel that remains intact atsea even though every original board has been replaced. FIG: REBUILDINGSCIENCE WHILE AT SEA He likewiserejected the rigidified semantics upon which suchstandard apriorisms are founded as rooted in anunscientific “myth of the mental museum.”31 Unfortunately, Quine’s writings rely upon asecond unfortunate form of semantic rigiditytracing to his Theory T conceptions of scientificmethodology, a theme to which we’ll returnwhen we consider the rationale that Ted Sidersupplies for the analytic metaphysics enterprise. As a result, Quine does not supply a plausiblydetailed account of why apriorisms typicallybreak down as science progresses. When Gauss complained about the blinkerednecessitarian thinking of his own day, he suggests a better answer:

But, methinks, in spite of the meaningless Word-Wisdom of the

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metaphysicians, we know too little or nothing at all about the realmeaning of space to stamp anything appearing unnatural to us as AbsolutelyImpossible.32

Here he criticizes a confident unwillingness to investigate the underlying utilities thatgeometrical talk provides on our behalf, including the question of whether reasoningrules that function ably locally (e.g., standard Euclidean inferential pattern) may failus globally (e.g., as applied to wider expanses of the universe). Gauss’s insights donot merely reflect the coarse conclusion that “the axioms governing space must beempirically established or supported by defeasible convention” (which is how Quineapproaches these issues), but stems from a deeper appreciation of the ways in whichinferential pattern within science pieces itself together into a successful patchwork.33

In this way, Gauss-like thinking reveals, through the formal developmentspioneered by his student Riemann, a deeper understanding than Quine’s of why earlya priori verities often collapse under the withering eye of the close scrutiny ofinferential policy. Operating in Essay 1's fashion as a term of mixed descriptive andmanagement import, the word “cause” tags along with the architectural decisions wemake in adapting an established strand of parent reasoning to engender strategicallymodified novel sons and daughters. Borrowing terminology from the mathematicians,we can say that the newer employments represent natural continuations34 of the olderusages. Often these continuations strike us as unique and completely unforced, withno evident whiff of empirical revision or conventional readjustment about them. Andthese same unnoticed developmental processes sometimes led to semanticcircumstances like some strange Twilight Zone episode in which an innocent hikerwalks through a woods that appears perfectly normal but is actually founded upon anon-orientable manifold akin to a Klein bottle. Our wayfarer blithely follows thepaths of natural continuation ahead of her, but somehow finds herself within a mysteriously altered landscape when she returns to her original position. Once theobscuring forestry is exfoliated, we discover the Riemann-like reason for herpuzzlement: an anholonomy (= topological twist) exists within the underlying surface. FIG: INABILITY TO SEE A MULTI-VALUEDNESS Entirely through innocuousprocesses of natural linguistic enlargement, seeds planted within our early aprioritraining can readily grow into incompatibilities of application that prevent analyticmetaphysicians from uncovering anything permanent in our usage other than faultynaive misrepresentations and empty platitudes.

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inability to see a multi-valuedness for the trees

(vi)

By failing to recognize the mixture of descriptive and architectural tasks that

“cause” commonly performs on our behalf, analytic metaphysicians wind up reversingcommonsensical conceptual priorities in strange ways. Here I have in mind simpleadvice such as “empirically map out the dominant causal relationships between layersin a rock through a microscope before you attempt to formulate predictive ‘laws’ forthe material.” These reversals can be clearly witnessed within the dismissivecriticisms that Jim Woodward’s work has received from partisans of the newmetaphysical school. Woodward has ably drawn for our benefit extensive maps ofthe multifarious places in which manipulationist questions35 play central roles withinthe developing architecture of successful science. Such studies are nicelycomplimentary to my own, which stem mainly from formal considerations of anapplied mathematics character (from which literature I have liberally lifted most of my“insights”). Woodward’s work, in contrast, arise from direct methodologicalobservation and encompass many forms of explanatory landscape requiring a greaterrange of diagnostic tools than I have managed to isolate. However, it is comforting tofind that Woodward’s work and my own dovetail rather nicely over the commonground we share. I anticipate that better diagnostic work within both methodologyand the mysteries of historical development within science will emerge as otherscholars learn to detect the telltale symptoms of manipulationist consideration. In light of these methodological virtues, I find it depressing that analyticmetaphysicians commonly dismiss Woodward’s work as trivial, beside the point orcircular. “Trivial,” because his explications don’t penetrate to the robust heart of thecausal processes we directly witness when a cataract of water careens along amountain hillside. “Beside the point,” because we employ “cause” fruitfully incontexts that plainly do not fit Woodward’s manipulationist emphases at all.

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Lewis-Stalnaker picture of counterfactual evaluation

“Circular,” because what might a “possible manipulation” comprise, except a specificform of counterfactual claim that needs “laws of nature” for its metaphysicalgrounding?

Let’s look more closely at these claims in-want-of-grounding, which take theform “If system S were manipulated in manner α, it would respond in manner β.” These claims are “counterfactual” in the sense that they report non-trivial facts aboutS even when S does not actually behave in manner α. Our metaphysicians complaintsabout grounding in the laws of nature arise from embracing some variant upon apoint-of-view articulated by Nelson Goodman and developed further by David Lewis,Robert Stalnaker and many others.36 Consider some counterfactual--the stockillustration is “if this match were struck, it would light”--whose antecedent specifiessome altered condition A* (“the match is struck”) that contrasts with the match’s realworld circumstances A (“the match is notstruck”). FIG: LEWIS-STALNAKERPICTURE OF COUNTERFACTUALEVALUATION Build a new set ofinitial conditions around A* in a“cotenable manner” (that is, adjust A*’sauxiliary conditions as close to those ofA as we consistently can) and thendetermine whether our laws of naturecarry us from these adjusted conditionsto B (“the match lights”) or not. If so,the counterfactual claim A B shouldbe regarded as true; otherwise not. Goodman despaired whether the cotenability criteria we employ to construct A*’sinitial conditions from those of A could be rendered pellucid, but Lewis and Stalnakerposited that this transition comprises a modal primitive--we natively understand howto evaluate the “closeness” of one possible world to another. Such assumptions allowthem to set up a logic for reasoning with counterfactuals (the details differ slightlybetween the two authors), which seems a useful enterprise even if we remainsomewhat in the dark about the underlying cotenability judgements. In the technicaljargon they favor, our very understanding of counterfactual constructions demands a“grounding” of a Goodman-Stalnaker-Lewis stripe founded upon a prior conceptionof law of nature.37

Upon this basis, Lewis then proposes a one-size-fits-all analysis of causal talk,whose details are less widely accepted than the portrait of counterfactual evaluation

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just sketched. However, most of his analytical confederates agree that our everydaycausal claims must be somehow established upon a metaphysically deeper footingthan Woodward’s counterfactuals offer, due to the “grounding” considerations of theprevious paragraph.

Something must have gone amiss in this reasoning because, in Examples I andII above, the primary motive for invoking a salient set of causal claims, articulated asan appropriate set of Woodwardian conditionals, is precisely one of skirtingdependency upon laws of the hypothesized sort. They serve as the chiefinstrumentalities of physics avoidance policies that exploit firm pieces of higher scaleinformation as a means of minimizing the quantity of unreliable speculation within ourphysical models.38 can. Thus the recognized genius of applying Lagrange’s methods,familiar to every practical engineer, to our seesaws (Example I) lies precisely in thefact that his stratagems allow us to ignore all of the forces (and their largely unknownattendant “laws”) that hold its sundry rods and springs together. Indeed, only doeslongstanding classical mechanical tradition fail to supply us with plausible “laws”governing the cohesion of rods and such, it most likely cannot.39 Almost In an alliedvein, straightforward attempts to supply reasonable estimates of the tensile strength ofgranite or pumice (Example II) from “laws” operating at the molecular level havepersistently failed to supply good results, whereas the policy of supplementing lowerscale consideration with upper scale structural data in the manner of our multiscalarmethods has enormously improved our ability to simulate complex solids oncomputers with confidence. Through such “law avoiding” procedures practicalscience gains considerably in trustworthiness. As other essays have emphasized,reliability remains a virtue undervalued within contemporary philosophy, but itsdedicated cultivation is one of the vital factors that distinguishes science proper frompseudo-science. A casual inspection of Martin Gardner’s In the Name of Science40

reveals that cranks can posit Theory T’s as ably as the rest of us, yet characteristicallycare not a whit about avoiding excessive speculation within their unreliable schemes.

The Theory T thinking still popular within philosophical circles today washesout the distinctive features of a clever reasoning architecture by employing fuzzyclassificatory categories (“auxiliary condition,” “law of nature”) in an unhelpfulway.41 Goodman and company’s inadequate understanding of counterfactuals tracesdirectly to this source, for they patently collapse significantly distinct forms ofexplanatory structure into the special format that mathematicians categorize as aninitial value problem, whose salient modeling equations possess an evolutionarycharacter (viz., they are of hyperbolic signature able to advance a target systemforward in time). They likewise presume that the equation’s annexed set of “initial

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a fellow scientismist

conditions”42 report the facts available on a single time-slice of the entire universe,chosen at whatever time the counterfactual antecedent A* is asked to be true.43 Conventional boundary conditions are not mentioned at all, let alone the constraintsand inter-scalar homogenizations that frequently serve as popular vehicles forreliability-enhancing physics avoidance (these two formal intercessions in explanatorypattern are central within Examples I and II). These descriptive limitations of theStalnaker-Lewis account are evident from the illustration provided.

Through succumbing to these “reduce every explanatory setting to evolutionarymodeling circumstances” propensities, the analytic metaphysicians overlook theobvious fact that counterfactual claims make perfectly good sense within explanatorycircumstances that are strongly equilibrium-centered or eschew any directconsideration of temporal consideration through other means. Our Example I seesawsillustrate this observation beautifully, for Lagrange’s descriptive tactics codify thehigher scale, constraint-based partial knowledge utilized precisely in the form ofsomewhat odd, Woodward-style conditionals:

If child i is moved through a vertical distance δ in a virtual manner, thismanipulation will cause X amount of opposing work to arise as a reaction onbehalf of the rest of the system.

As Essay 7 emphasizes, we want to collect these individual assertions into a“possibility space” that is ample enough to guide our reasonings through dense sets ofimproving approximations to reach the desired equilibriums as fixed points, but whichmust also exclude extraneous counterfactual junk that would cloud the clarity of thesewell-marked pathways. Upon what epistemological grounds do we frame theseappropriate spaces of counterfactuals? Answer: through the direct inductiveenlargement of experimental results (poking the apparatus at various points, includingthe locations of the children), familiarity with the fact that beams rarely flex or breakunder small, child-like loads and a dash of mild “theoretical” enlargement.44 In whatreasonable sense of “grounding” must such claims be “grounded in laws,” when theirentire raison d’être is one of avoiding undue speculations of thatnature?

In truth, I am at a loss to understand what philosophersseek when they demand a grounding for counterfactuals beyondthe fact that they have become captivated by the formaldistortions of the Stalnaker-Lewis picture. Sometimes, itappears, they merely wish to endorse some amorphous form ofscientific realism (e.g., “the entire universe is run by physics”). Well, I’m as hard-bitten a scientismist (= practitioner of

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scientism) as walks the earth, so I would cheerfully acquiesce in these wuzzysentiments were I to understand them. FIG: A FELLOW SCIENTISMIST But suchharmless proclivities supply no excuse for ignoring the fact that real life physics, aswe now have it, actively exploits constraints, multiscalar linkages, paysthermodynamic attention to controlling variables and much else as part of its dailyroutine. And we’ve just seen that Woodward’s counterfactuals form an integral partof the architecture in which those strategies get implemented. So it is disheartening tofind philosophers airily dismissing Woodward’s work as “irrelevant to themetaphysics of ‘cause,’” when it should instead serve as a beacon casting a revealinglight upon the foggy inadequacies of Theory T conceptions of scientific procedure.

Sometimes, if one follows Paul’s centering of metaphysical endeavor within theprerequisites of early apriori training, talk of grounding can be converted intoconsiderations of learning dependency. But such considerations do not support theStalnaker-Lewis view at all. As children playing on seesaws, we soon learn how toenlarge a set of experimental observations into dense sets of possible observations(albeit not of a “virtual” cast), long before we entertain any firm notion of a “law ofnature’” (at least of a sort we can distinguish from “Mommy’s rules”). Indeed, thefocused spaces of possibilities we assemble in this manner are distinguished by thefact that they directly suggest the little steps in reasoning we should take in addressingbasic questions about our seesaws, e.g., “if I jump off suddenly, what will happen toJack?” Surely, one of the crucial skills we must acquire in our early apriori training isan ability to arrange our thinking within an appropriately focused arena for reasoning,in the manner “let’s concentrate upon this specific set of clues and see where theylead.” Mentally engulfing a seesaw S within a swarm of reliable Woodwardiancounterfactuals “if S is tweaked in manner α, it will respond in manner β” provides allof us, children on the playground included, with a popular framework for guiding theheuristics of our reasoning onward in a useful manner. 45 But acquiring basicreasoning hygiene of this focused character strikes me as a quite different process,psychologically, from learning to pick out “laws of nature” (insofar as we ever masterthat purported skill at all).

Allied criticisms apply to popular hand-waving to the effect that counterfactualclaims “require truth-makers,” which can be variously interpreted either in the fuzzy,but innocuous, physics-admiring fashion mentioned previously, or as a rigidifiedsemantics thesis as wildly implausible as the contention that children learn “law ofnature” before they master counterfactual constructions. In particular, the assumptionseems to be that “requiring truth-makers” should be understood as a semantic demandfor a recursive dependency story in the mode that Tarski supplies for the logical

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connectives (“A & B” is true if and only if A is true and B is true), where “law” issupposed to show up at some elemental stage within these recursions. In terms ofpedagogical reality, this etiology is obviously fictitious, for we master the rudimentsof counterfactual construction, causation, manipulation, etc. in one big initial jumblewhich we learn to adjust to more refined purposes later on.

Indeed, our seesaws discussion effectively illustrates the sense in which we canconfidently assign permanent truth values to a collection of statements without anyproper understanding of the supportive underpinnings that supply these sentences withtheir information-conveying merits. Today we recognize that Lagrange’s techniques,and his allied spaces of possibility, allow us to isolate the capacities for potentialenergy storage latent within a piece of connected apparatus, although that the relevantsupportive concepts were not cleanly articulated until well into the nineteenth century. As examples throughout these essays repeatedly underscore, we often mastereffective reasoning routines long before we properly appreciate their true strategicunderpinnings. If we can perform successful reasoning feats repeatedly, then, in theadvice of one of my favorite mottos: “a method which leads to true results must haveits logic.”46 In the present context, such considerations direct us to search for thesecret and often complex manner in which the sentences employed within Lagrange-style reasonings encode the requisite physical information to carry us reliably to “trueresults.” Sometimes, after many years of patient research, we find ourselves able toanswer such questions with empirical confidence. But this sort of informational“grounding” doesn’t appear to be what our analytic metaphysicians seek and certainlyfails to conform to the visions of armchair certitude that “‘A & B’ is true if and only ifA is true and B is true” excites within apriorist minds. Put another way, if the“philosophical theories of truth” we articulate can’t supply a natural sense in whichLagrange’s manipulationist counterfactuals come out as unproblematically true, somuch the worse for these “philosophical theories of truth.”

Occasions where effective reasoning methods are first encountered in the guiseof spaces of directive counterfactuals are not uncommon in the history of science andthis observation underscores the importance of Woodward’s work as a diagnostic toolwithin philosophy. One of my chief objectives of this book is to provide plausiblenarratives of why conceptual confusions often emerge as the collateral effects ofsubtle but otherwise healthy patterns of descriptive improvement. In the appendix Iillustrate how close attention to several layers of experiment-linked Woodward-styleconditionals explain the emergence of common muddles with respect to “pressure.”

(vii)

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Throughout the preceding discussion, I have expressed an enduring sympathywith L.A. Paul’s emphasis upon what we have called “early apriori “ considerations. In my estimation, her most pungent observations with respect to the prerequisites ofconceptual developmental interlace nicely with the findings of modern childpsychology and should be understood in that light. My main objection to her defenseof analytic metaphysical doctrine per se lies in the fact that the Neurath’s boat-likedemands of improving descriptive practice must eventually tear our inferentialdoctrines away from the simpler piers at which they were originally docked.

I do not view the methodological suggestions of Ted Sider with equalindulgence, for they spring directly from the Theory T misapprehensions with whichthese essays are collectively concerned. As I understand Sider, he anticipates that,ultimately, the detailed ontology of the universe must be specified by science, but theadept metaphysician can anticipate its basic contours ably enough to get with thephilosophical work right now. And how does he know these “basic contours”? Helearns them from his Theory T philosophy primers. In making these presumptions,Sider must know what the syntax of some “fundamental theory” yet to come will looklike, after it is freed from all the delicate interweavings of constraints,boundary/interior harmonization worries, multiscalar dependencies, reliance upondifferential equations, etc. that mar the landscape of working science today. Why theneed for futuristic appeal? Because at the heart of Sider’s enterprises are a range ofdistinctions–-“perfectly natural properties,” “internal versus relational properties,“counterfactuals sustained by explicitly articulated laws”–that can only be “read off”a “fundamental theory” of pristine contours, of a canonical Theory T ilk. We haveobserved that the long career of classical mechanics is characterized by its stolidunwillingness to suggest plausible “laws” for basic material cohesion, usuallypreferring the more reliable, if metaphysically unenlightening, dodge of relying uponconstraints and allied evasive crutches. In a similar way, the complicated forms ofdescriptive cooperation documented in these essays render clean distinctions between“internal and relational properties” dubious, insofar as I can determine. I’ll comeback to the business of “perfectly natural properties” in a bit.

In accepting such views, philosophers have succumbed to what I call a “manproposes; nature disposes” view of the theoretical enterprise--the notion that we canpretend that complete theories spring fully formed from the brows of their creators. Availing ourselves of this pretext, they claim that our philosophical musings willcome to no consequent harm by crediting our scientific doctrines with the full logicistregalia of Theory T thinking, even if real life science prefers to evolve more carefullyand slowly, leaning upon impure injections of constraints etc. to steady its onward

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march. But I disagree; through such allegedly “harmless idealizations” we erase thearchitectural complexities at the root of much that is puzzling within the progress ofscience.47

An invidious second form of semantic distortion creeps onto the stage in thecompany of this imposition of Theory T structure: the presumption that suitablyarticulated theories “implicitly fix the meanings” of their specialized vocabularies. This doctrine represents an old conceit within analytic philosophy, dating to the turnof the twentieth century. The purpose for which the dogma was originally devised isnoble enough, supplying a rationale for rejecting the stifling restrictions onspeculation that earlier methodologists had extracted from their “metaphysically”conceived canons of causation or their traditionalist standards of conceptual clarity.Some of this pre-history is recounted in Essay 4 and, at much greater length, in WS. Against these restrictions, Theory T thinking supplies an environment in which the“free creativity of the physicist” can freely blossom. To do this, the logical contoursof a freely creative T must be firmly specified, in David Hilbert’s axiomatic fashion,allowing the deductive instrumentalities of T to endow its theoretical sub-vocabularieswith “meanings” entirely derived from T’s formal specifications. Though suchappeals, the conservative cavils of stuffy metaphysicians could be efficientlysidestepped.

Well, these liberationist motives are entirely commendable, but the demandsplaced upon the theoretizing intellect are excessive and too unreliable to be satisfiedwithin a plausible scientific practice. We scarcely want to nail the boards of our shipof science so tightly together in “man proposes; nature disposes” fashion that wecan’t replace a dubious timber when some dilemma of mathematical disharmony popsup in the manners documented in these essays. Questions of the “meaning” ofscientific vocabularies must be addressed in a subtler manner than “implicitdefinability” permits.

In the fullness of time, these logic-centered proposals have gradually hardenedinto firm dogma amongst a large array of philosophers, leading to a situation wherefew analytic metaphysicians reveal any appetite for attending to the complications ofreal life scientific architecture. Like Ted Sider, they are content to appeal to ahypothetical future “fundamental science,” where all of today’s distracting cobwebshave been swept away in favor of doctrines of pristine Theory T contours. Basedupon conversations I have had, few members of this new metaphysical schoolrecognize how many substantive questions with respect to the representationalcapacities of applied mathematics become suppressed through blithe idealizations of afuturist cast. In contrast, many critics (myself included) believe that proper

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definable and important traits

metaphysical concern should focus exactly upon the troublesome issues ofrepresentational capacity that become suppressed through these means. I’ll return tothese issues of “proper metaphysical concern” in our final section.

(viii)

Many of Sider’s enterprises depend upon his ability to sort out the “perfectlynatural properties” of the world, following David Lewis in his “New Work for theTheory of Universals”48 (a founding document of the analytic metaphysics school). These traits, Sider believes, are the ones that properly “carve reality at the joints” andwill be concretely delineated within the syntax of his hypothetical final theory. Trueto form, he doesn’t much care what this theory will be like in detail; he merely wantsto know the general classificatory categories into which its terminologies will fall. Upon this basis, he believes that he can grade the collection of predicates we mightrecursively assemble from the vocabulary of T into a priori categories, viz., as directlyphysically descriptive (“the force between these planets obeys Newton’s Gm1m2/r2

rule”), semi-artifactual (“the planet has a mass of 1024 kg measured within the metricsystem”), non-projectable (“the planet has mass 1024 kg if observed before the year3000 and 103 kg otherwise”) and altogether useless (“the planet travels through theposition (7, -8 + 6i, -37i).” Through these distinctions, the “kind terms” of a scienceallegedly fall into tidy orbits delineated by their grammatical construction.49 FIG:DEFINABLE VERSUS IMPORTANT TRAITS

Such fixed vocabulary views rest upon the popular propensity of allowingexperience in elementary logic class to serve as our central paradigm of scientificorganization. However, if we work with differential equations within our science, weare forced to look upon “important classificatory concepts” in a far different manner. This is because such equations grow many of their importantconcomitant quantities from infinitesimal seeds; they are notdirectly anticipated within the subject’s preassigned syntax. The Fourier-style characteristics of an uneven string exhibit thegeneral pattern. In terms of practical importance, thesedecompositional traits capture the basic behaviors of our stringin the most direct and efficient fashion, but their specificationsare rarely captured within the recursive orbit of thevocabularies we employ in setting up their differential equation seeds.50 Instead, theyemerge as the fixed point limits of holistic approximations, whose existences must beestablished by set theoretic means. Whether or not such “limits” truly exist is often a

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lifting reasoning from D to D*

descriptive over-extension in a differential equation model

delicate matter. Many philosophers, Sider included, further

presume that inferential activity within sciencetranspires within some restricted grammatical arenacarved out by the “logical forms” of its axioms(which fall at, e.g., a Π1

3 level within the so-calledanalytical hierarchy). But this is not so; a physicistis apt to wander far afield in GreaterMathematicsland in search of a salient purchase upon the significant traits of hertarget system. Throughout these essays, we often consider the inferential advantagesthat attach to a factoring, viz., uncovering a set of descriptive parameters that allow usto decompose complicated behaviors into simpler sub-behaviors. Much of our abilityto reason effectively about the natural numbers traces to our ability to factor themuniquely into primes. This advantage is so pronounced that mathematicians generallyattempt to regain these virtues even when studying domains D that lack the property(e.g., the algebraic ring of15). How do they do this? Through embedding D within aricher domain D* to which justenough extra numbers havebeen added that uniquefactorization is restored. Thehope is to harness the “easyproof” capacities of D* to findtheorems that can hopefully bepulled back down into the original D. FIG: LIFTING REASONING FROM D TOD* If this strategic gambit proves successful, it allows us to pull down the propertiesuncovered in D* and apply them meaningfully to the original membership of Dproper. Accordingly, the traits we encounter within D* often supply deepercharacteristics of our target system than are syntactically constructible within in theoriginal vocabularies parochial to D.

Indeed, the differential equation seeds from which our string’s modal propertiesgrow are rarely as descriptively reliable as the fruits they produce. As the discussionin Essay 5 illustrates, we usually obtain our differential equations models byartificially extending higher dimension scaling rules down to an infinitesimal level,even though we are fully aware that these behavioral assumptions palpably fail at thelower size scales.51 FIG: DESCRIPTIVE OVER-EXTENSION IN A

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DIFFERENTIAL EQUATION MODEL Accordingly, we usually don’t regard thedifferential equations themselves as supplying directly descriptive accounts ofphysical behaviors below the cut off level where the scaling assumptions fail, andinstead regard the formulas merely as convenient way stations to pass through as wesearch for more trustworthy conclusions. In contrast, the Fourier-like modes obtainedfrom these infinitesimal seeds represent some of our target’s systems most reliablecharacteristics, captured upon a macroscopic, dominant behavior basis. In thesame vein, philosophers also presume that a theory’s “most fundamental traits” arethose required to express its basic “laws.” But our string’s modal properties do notobtain their status as “important trait” from laws or differential equations alone, butfrom the ways in which these interior considerations hook up with their side conditioncompanions. The latter not only include boundary conditions, interfaces and basicmodeling assumptions, but also the unformalized appeals to dissipation, relaxationtimes and steady state condition we have discussed in other essays. In particular, ourFourier string modes gain their descriptive centrality from the fact that the string’sendpoints continually redirect traveling wave energy back into the interior, coupledwith a bit of relaxation time dispersion. These boundary condition behaviorsencapsulate completely different physical factors than are registered within theinterior string equation, but without their assistance the string loses its capacities forstoring energy through standing wave characteristics.52 Understanding how thesecomplex forms of cooperative partnership operate is essential to a reasonableassessment of how science manages to capture the physical world within its appliedmathematics snares, so whitewashing these delicate harmonizations under thick layersof Logic 101 paint is not helpful.

Essay 3 discusses how intimately Leibniz’ metaphysics is entangled with thedescriptive overreach of differential equations just mentioned. He concludes thatsuch equations cannot directly reflect physical reality in a naive way, but must beparsed in a manner that accurately recognizes the expressive limitations of the tools ofapplied mathematics. The metaphysical entities he reaches via these reflections arehis strange monads. None of us today are likely to accept these Leibnizian remedies,but the interpretative problems that drive him to these extremes remain with us still(and, indeed, have become more complex). Right now, we cannot extract a plausible“ontology” from physical doctrine in the naive syntactic manner suggested by Quinein “On What There Is.”53 Theory T brainwashing had persuaded him that reasoningwithin mathematical physics operates very much like what it does within Logic 101,only with more numbers. But this is wrong and we shouldn’t continue to followQuine’s logic-addled policies. I don’t have crisp counter-proposals to offer on this

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pristine and perturbed landscapes

“how to read the book of science” score, except to follow the suggestions that plaincommonsense offers when confronted with clear examples, such as the stringbehaviors just surveyed. Quite possibly descriptive policies within appliedmathematics have not stabilized adequately enough that more systematic suggestionscan be plausibly offered at the present moment .

Such considerations reveal that Sider’sfuturist theory T must operate mathematically in amanner different from today’s, for the basicequations of his hypothesized T must characterizeits universes in an utterly straightforward way,working, presumably, from the infinitesimal levelupwards. It’s certainly true that physiciststhemselves sometimes talk of some hypotheticalset of equations that will fully govern the entireuniverse in such a totally unlayered way. Some ofthis merely constitutes an exaggerated looseness ofexpression,54 but some of it traces to underlyingassumptions much like Sider’s. I don’t think thatphilosophy as a discipline should latch its wagonunreflectively to such prophecies; as much of ourdescriptive task lies in tallying up what we don’tyet know, as expounding upon the firmlyestablished. Nor do I see the percentages inwagering on a horse race in which all of the nagswill cross the finish line long after we are dead. .

(ix)

Some further remarks on the “fixed orbit” conception of physical traits mayprove useful. The Fourier analysis55 we applied to our string is a factoringtechnique–-its overtone spectrum allows us to reason about our string’s behavior is avery effective way. But what relationships must these factoring quantities bear to thetarget physical system under scrutiny? Let’s first think about how we typicallycategorize topographical features in real life

Scenario 1: Suppose that we encounter a landscape containing two improbablyperfectly conical volcanos. FIG: PRISTINE AND PERTURBED LANDSCAPES One readily appreciates that the most computationally convenient policy of describing

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a suitable Book of the World

this tableau is to employ three different coordinate maps as follows: local map A,which lays down conical coordinates with their origin at the apex of volcano A; localmap B, which follows the same policy with respect to volcano B and global map C, aCartesian style rectilinear map that locates the two volcanos on a common map wheredistances between them can be readily calculated. Why is this particular partitioningof descriptive practice advisable? Answer: because of the way that volcanos formand the importance of elevation with respect to plant life and soil chemistry. Suchdistributions are likely to be pretty much the same at the same radial distances rA andrB from the peaks, whereas a location’s angular location θA or θB will display no alliedregularities whatsoever. This independence allows us to reason about vegetation andsoil solely in terms of a location’s rA or rB coordinates values, ignoring thebehaviorally irrelevant θA or θB values.56

Scenario 2: Suppose we instead consider a more uneven landscape whosetwinned mountains are less perfectly conical due to the influence of various perturbingfactors. As long as the degree of disturbance is not too great,it will still be advantageous to utilize the same conicalcoordinates, despite the fact that they no longer preciselycoincide with the physical boundaries of the terrain. And oursimple rules for vegetation and soil will still capture the centralfacts about the terrain (“to first order”), allowing us tointroduce corrective factors as perturbations upon the simplerdistribution patterns. The continued algorithmic advantages ofradial factoring persists even within this messier situation. Indeed, the vibrational modes of any probable physical systemare like this, because small non-linear perturbations foreverspoil the perfect factoring of the linear string. Typically wedesignate to these perturbed modes in terms of their closest“perfect factoring” cousins: this allows us to state (in a non-contradictory way) that “because of leakage between the modes, the 440 frequencymode in this string actually cycles imperfectly at a rate closer to 436 cycles persecond.” Quine called linguistic strategies like this “deferred ostension”--weindirectly draw a listener’s attention to A by pointing directly at its more perfectedcousin B.

This innocuous observation is worth making because respondents havesometimes objected to my “many important quantities lie outside the grammaticalorbit of a theory” observations by retorting, “What do you mean ‘they can’t benamed’? You’ve just named them.” Yes, through deferred ostension by employing

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language extracted from Greater Mathematicsland, rather than anything specific to theTheory T under consideration. If one tolerates such liberal conception of“definability,” then the most recondite ontologies of modern physics can be readilydefined within the orbit of your favorite comic book.57 FIG: A SUITABLE BOOKOF THE WORLD

Even within the messier contours of Scenario 2, we still want to “see,” as itwere, the perfect conical volcanos embedded in the landscape, for, even within thesemuffled circumstances, the conical coordinates continue to capture the dominantbehaviors on display, albeit hidden beneath an obscurant layer of rocky complication. Although the distribution of hillside vegetation will become somewhat messier due tothis increased cragginess, distance from peak still remains an effective descriptiveparameter we want to employ centrally in our reasonings, even if the “peak” to whichwe refer can no longer be witnessed in the volcanos before us, having been blownaway in some ancient explosion of volcanic gas.

One of the reasons why I strenuously object--I’d ban the phrase from politecompany if I could--to loose talk that “science always idealizes” is that oftentimes theassertion signifies nothing more than “in attempting to reason profitably about acomplicated topography, first find a structurally related landscape where simplereasoning procedures R work well and then try to perturb these R rules to suit thecircumstances before one.” This is simply how prudent investigative strategy unfoldswithin most areas of mathematics (indeed, within reasoning more generally) but haslittle connection with the more pernicious “essential idealization” themes that oftenparade under the same “science always idealizes” banner.

Scenario 3: In the same spirit, we can hold onto an underlying factoringdescription even when we accept significant regions in which the dominant behaviorsallied to the factoring completely disappear. This occurs, for example, if the valleyregion between volcanos A and B becomes filled with fallen rubble, creating aboundary region in which the local soil and vegetation fail to conform to our “distancefrom mountain peak” rules at all (mathematically, the clean seperatrix that segregatesdomain A from domain B in scenario 1 has enlarged into what is sometimes called a“mushy region”). A common policy for handling these intermediate dominions todevise a simple rule that smoothly interpolates between A and B regions, based upontheir separation upon a common map. Often these fill-in rules are as coarselyarticulated as a standard lower dimensional boundary condition, with little claim todetailed descriptive accuracy, for their primary role is to allow us to connect theMountain A reasoning pattern to that pertinent to Mountain B.58

Although I have here stressed the “importance” of a physical quantity in terms

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Helmholtz envelope for a single traveling wave

of its suitability as a vehicle for effective reasoning (which is the manner in which Iwould convert Sider’s alleged “projectability” into a defensible linguistic category),we have also observed that physical considerations of manipulation and control playcognate roles in establishing the dominance of a particular quantity on a size scalewithin nature. As we noted, the chief difference between a strain energy and anincreased temperature lies in the fact that bulky entities on our scale size can’t employthe latter as efficiently for storing coherent work capacity. These limitations in turnrecommend a linguistic strategy of designating the highly controllable and lesscontrollable modes of energy storage underdifferent headings, even though these divisionsresemble the dominant behaviors of our Scenario 2volcanos in not being perfectly demarcated withinnature.

In fact, these issues of “manipulationrelative to a scale size” sometimes justify thesalience of a decompositional description evenwhen the target system itself doesn’t fullyrationalize the policy. As I’ve acknowledged invarious footnote retractions along the way, the trueenergies of bowed violin strings do not subside into standing wave modal registers asrapidly as I (and most textbooks) pretend, but retain more of their original travelingwave character over the duration of the tone. What instead occurs is that theinstrument body and the intervening air respond to the string’s twitchings in acomparatively sluggish fashion, enchained to the Helmholtz envelope carved out bythe string’s rapid wigglings, rather than directly reflecting its wavelike movements(only an exaggerated envelope for a single traveling wave is illustrated, but thecontours appropriate to normal violin tone will consist in a large number of smallersuperimposed patterns of this ilk). FIG: HELMHOLTZ ENVELOPE Accordingly,the “440 cycle response” of a violin string is secretly cooperative in character; itrepresents an ability to induce its environment to store energy within a 440 cyclemode.

Essay 5 emphasizes the important role that homogenization and alliedtechniques play in supplying mathematical surrogates for the environmental andinterscalar filtering that play major roles in how natural behaviors piece themselvestogether. None of these considerations play any discernable role in Sider’s approachto the “kind terms” of science. But I believe that his loose appeals to distinctionsbetween “internal” and “relational” properties need to be more carefully scrutinized in

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Galileo

light of these cooperation-of-linguistic-labor considerations. A linguistic lesson taught here is that the syntactic labels we assign tosignificant physical behaviors often derive from the mathematical terrain in which theadvantages of strategies like factoring emerge as clearly registered (e.g., within theextension ring to -15, rather the original set of algebraic numbers), rather thanappearing within any grammatical orbits directly spawned by the original differentialequation modelings.

Finally, the fixed orbit point of view overlooks one of the vital functions of settheory within applied mathematics: repairing the disharmonies that commonly arisewhen we attempt to persuade our natural choices of descriptive tools to work togetherin a mutually supportive fashion. In a nice stroke of unintended irony, Sider extractsthe title of his chief book from a famous quotation by Galileo in The Assayer:

Philosophy is written in this grand book — I mean theuniverse — which stands continually open to our gaze, but itcannot be understood unless one first learns to comprehendthe language and interpret the characters in which it iswritten. It is written in the language of mathematics, and itscharacters are triangles, circles, and other geometricalfigures, without which it is humanly impossible to understanda single word of it; without these, one is wandering around ina dark labyrinth.59 FIG: GALILEO

In fact, Galileo’s triangles and rectangles have long posed a non-trivial set ofharmonization headaches upon mathematical physics, because boundary regionascriptions of this geometrical character do not accord well with the differentialcalculus tools we employ to describe the interiors of triangle-like things. Why? Because the sharp corners can concentrate stresses in such a way that they “blow up”in a manner that undercuts the validity of the interior equation. On the other hand,these same perfect triangles also facilitate a wide variety of fruitful reasoningpractices that become blocked if we prudishly decide that only figures with roundedcorners can be “found in nature.” Today applied mathematicians engage in variousfancy forms of functional analysis footwork to impose an improved cooperativefamily behavior upon Galileo’s insufficiently sophisticated triangles. Thesetechniques are discussed at some length in Essay 8.

In other words, in invoking “triangle” talk as they naively did beforemathematicians developed the functional analysis corrections in the 1950s, olderscientists genuinely “wandered around a dark labyrinth” in the sense of that they werehiking through a deceptive semantic landscape, whose Twilight Zone complexities

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they could not discern due to the obscuring forestry.In short, fixed-orbit-of-quantities theses seriously underestimate the

complexities that applied mathematical thinking confronts in adapting its descriptivetools to suit a wider range of applicational circumstances. These concerns are neverstatic–we never set up “all the applied mathematics we will need” at some fictitiousmoment of “man proposes; nature disposes” postulation. Rather we find ourselvescontinually confronted with situations in which nature’s unfolding processes fall outof step with the pathways of reasoning we have laid down within our previousmathematics. Correcting these discrepancies generally requires some sophisticatedform of ameliorating repair, often drawn from exotic corners of GreaterMathematicsland. Theory T obliviousness to these enterprises of cooperative repairengenders a static conception of “ontology” wherein the metaphysician believe thatshe can intelligibly ask, “Is it possible, through some sort of reductionist trick, toeliminate our prima facie reliance on “abstract objects” such as numbers in favor ofthe purely physical?” Such misbegotten “metaphysical projects” rest upon asimplistic portrait of how science and mathematics advance in tandem, of the sortdiscussed at greater length in Essay 9.

To summarize a long discussion: Paul’s emphasis on early apriori learningproperly directs our attention to the linguistic question of how we talk about andmanage a wide variety of differently strategized explanatory schemes. Ourdisagreements with Paul lie largely with the rigidified semantics assumptions shemakes in presuming that words like “cause” will retain a constant, metaphysicallyanalyzable “meaning” throughout all of its helpful ministrations. This is a premisethat I reject, in company with Quine and other critics of permanent necessity.

Sider, on the other hand, makes a wide range of standard Theory Tassumptions about how the predicates within a posited science sort themselves outwith respect to “importance.” To me, his static point of view locks science within acoarse framework that fails to recognize the distinctions required to understand themore subtle adjustments at the heart of its improving practices.

Somewhat artificially,60 I have divided analytic metaphysical opinion into thesetwo themes, because I find that their proponents often engage in dialectical bait-and-switch without realizing it. Specifically, if we challenge whether their early aprioritenets apply to working science today, they shift to futurist soothsaying: “Yes, currentdoctrine displays those defects but they will all be corrected in the great Theory T yetto come.” But we were given the impression that our early apriori metaphysicalwords provide the guideposts for all theorizing, including that pertaining to the tawdrypresent.

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our heros

(x)

Within their own panegyrics, the analyticschool portrays its endeavors as a return to the“metaphysics” of days of yore, triumphantlyrebounding after suffering through a twentiethcentury dark age in which these pursuits were squelched by puritanical positivistssuch as Carnap and Quine. Perhaps this self-assessment of heritage mightappropriately apply if we confine our attention to the Scholasticists of medieval times,whose suggestions the analytic metaphysicians often echo closely. But it does notjustly apply to the venerable authors (e.g., Descartes, Leibniz, Kant, Peirce orCassirer), who placed the puzzling policies of mathematicized physics near the centerof their philosophical concerns. FIG: OUR HEROS Descartes, for example, worriedthat our computational abilities to follow physical process through mathematicalmethods are limited and confined to special opportunities in which the physical worldhappens to fall into matching accord with the representational structures availablewithin mathematics. He believed that, as coarse finite beings, we are incapable offollowing the particles of a fluid as they pass through a sharp constriction in anydetail. He contended instead that they enter an opaque zone of inferential“indefiniteness” and that we can pick up the descriptive threads only on thedownstream side of the pipe, relying upon the conservation of bulk mass andmomentum across the intractable interior.61 In this pessimistic assessment, he washampered by not having the infinitesimal calculus available to him but, even withmodern resources, a certain degree of subtle cooperative family interdependence isrequired to carry a fluid system through the counterflow region of a sharp restriction. In a related manner, Leibniz’ metaphysical worries about the “labyrinth of thecontinuum” are closely associated with the descriptive overreach typical of calculus-based physical models–see Essay 3 for more on this. Allied illustrations can beoffered with respect to the other canonical metaphysicians listed.

Two recurrent themes in this book are: (1) rather subtle forms of mathematicalrepair are required to bring our reasoning tools into adequate accord with physicalphenomena and (2) concocting these repairs requires a good deal of subtle thinkingabout strategy and reference within language. For better and worse, these twinnedthemes comprise significant aspects of the “metaphysical” thinking encountered in thehistorical authors listed. Without recognizing that they have done so, their analyticsuccessors have swept most of these topics of traditional concern off their diningtable, leaving behind comparatively wan fare with respect to, e.g., whether “ontic

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how metaphysics is now treated

commitment to abstract objects” might be avoided throughengaging in strange policies of grammatical circumlocution. The latter, I submit, is not the stuff of which a sturdymetaphysics should be fashioned. This thinning out oftheme largely traces, I believe, to an excessive trust in thesoothing nostrums of rigidified semantics and Theory Tdoctrine. By following these doctrinal Pied Pipersuncritically, our metaphysicians reduce the puzzles of reallife inferential technique to puny dimensions, casting out

important metaphysical concerns along with the scientific, mathematical and linguisticwaters in which these topics properly bath. FIG: HOW METAPHYSICS IS NOWTREATED

In these respects, a passage from our earlier quotation from Ted Sider isrevealing: “A realistic picture of science leaves room for a metaphysics tempered byhumility.” What does he mean by “a realistic picture of science”? In light of theconsiderations raised in this book, surely not “an unbiased portrait of scientificmethod based upon a close observation of real life practice,” for then Sider would notdismiss Woodward’s descriptive studies so airily. Nor can he intend “from a non-instrumentalist point of view that maintains that science talks about a genuine externalworld,” for Descartes was surely a “realist” in this sense, but one who further claimedthat the inferential powers afforded to us within mathematics are limited and cannotkeep pace with the external world except in fits and starts. We have seen thatanalogous issues continue to remain a matter of methodological concern and shouldnot be lightly dismissed. No, what Sider actually means is “a picture of science thatentirely conforms to the Theory T percepts I have been taught, in which thereferences of its terminologies conform to simple Tarskian pattern.” In doing so, hepresumes, in the face of presumptive evidence to the contrary, that, upon our final dayof epistemological reckoning, the Almighty will present us with a tablet inscribed withan array of “laws” that fully accord, structurally, with every syntactic perceptanticipated by David Lewis in his writings on science. This prophetic viewpoint, Isubmit, does not qualify as “humility,” but reflects a philosopher’s presumption thathe or she can ably pronounce upon the “basic structures of science” without botheringto study them in any detail.62

From a formal point of view, Sider’s conception of metaphysics as a pre-scientific enterprise rests upon the syntactic conviction that his “final physics” willassuredly assume the structural contours of what Essay 2 formally characterizes as an“evolutionary physics of hyperbolic signature operating on a cosmological level.” Inthat rosy happenstance, all of today’s annoying cobwebs of scales that must “talk to”

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philosopher trying to be an ox

one another through roundabout filtering devices and of interiors and boundaries thatmust be harmonized with one another in unequal but complementary manners willhave entirely vanished, replaced by a resplendent theory T of impeccable first-orderlogic credentials. For it is only within such a admirably chastened future that Siderwill readily find the pristine “laws, properties, lasting through time” required toground his present day metaphysical speculations. I cheerfully acknowledge that sucha futuristic eventuality might come to pass–the affair ultimately lies in nature’s hands,not ours. But I discern no evident necessity within, nor strong inductive supportbehind, these expectations now and feel that contemporary philosophy will do betterif it concentrates more firmly on the here-and-now. For that project I heartilyrecommend the diagnostic value of Woodward’s manipulation conditionals.

I write these passages with faint positive expectation, for, in my personal trafficwith analytic metaphysicians of stout timber, I have been struck by their palpable lackof interest in, e.g., how the materials scientists down the hall resolve the puzzles thatmereologists set themselves with respect to boundaries and interiors. But perhapsI’ve merely had the misfortune of running into an especially dogmatic sampling,although I suspect that such dismissive propensities trace to more fundamentalsources, such as the two founts of doctrinal rigidity I have isolated. On the surface, atleast, the analytic metaphysics movement gives the impression of a creed that rebuffsthe counter-suggestions of empirical study as robustly as any other sect that promisesits members similar allotments of contentment and self-satisfaction.

In one of those cruel whimsies of philosophical fate, Quine’s Theory T-lacedsuggestions on ontological commitment in From a Logical Point of View havebecome central dogma within this new theology, in a mannerthat invokes entirely unQuinean visions of a kingdom ofnecessitarian doctrine in which a clever philosopher canluxuriate without knowing much of anything about anythingelse. As it happens, Quine’s book title derives from a classiccalypso composed by The Lion (“From a logical point ofview/Always marry a woman uglier than you”). In that samespirit of light irony and in tribute to Quine himself, I haveborrowed my own title from another of Lion’s evocativecompositions, this time satirizing the enthusiasms of religious cults:

Yes, believers! We come from the glory--we come from the gloriouskingdom.63

In present circumstances, the relevant “kingdom” represents the permanentenshrinement and glorification of the humble but genuine linguistic considerations,

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familiar to all, that I have labeled the early a priori. In the process, a legitimate froghas puffed itself up into an unsustainable ox. FIG: PHILOSOPHER TRYING TO BEAN OX

We should instead temper our ambitions of grandeur with the cautions ofOliver Wendell Holmes:

[L]ogical method and form flatter that longing for certainty and for reposewhich is in every human mind. But certainty generally is illusion, and reposeis not the destiny of man.64

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Appendix: The Origins of Conceptual Clashes I

In essence, the foregoing critique of analytical metaphysics follows the basiccontours of W.V. Quine’s well-known criticism of the analytic/synthetic distinction.65 Both arguments rely centrally upon the observation that normal scientific advancecharacteristically twists the original usages of words in manners that rarely leavebehind any necessitarian residue. However, Quine’s argumentation leaves him opento an effective retort, here associated with the suggestions of L.A. Paul. It works likethis. Quine’s own thinking suffers as much from the distortions of Theory Tpresumption as any of his opponents and he likewise assumes that everything we doin science falls under the umbrella of some grand, amalgamated Theory T of thischaracterization. “Not so fast,” our metaphysicians retort, “we surely requirelinguistic tools to assist us in adjusting our theoretical endeavors, as when we rejectan old T in favor of a replacement T’. But we’ve argued that the various vocabulariesof “cause” and “effect,” mereology and logical possibility serve as the pre-scientificguideposts we follow in theory constructions of this sort. Quine’s internal-to-sciencesemantic wobblings cannot affect the metaphysical categories we follow in framingfresh bodies of theory. Our metaphysical task is to catalog these fundamental pre-scientific invariants accurately.”66 In truth, I’ve always wondered how Quine reconciled his evolutionary semanticpragmatism, as epitomized within his Neurath’s boat metaphors, with the staticconception of “implicitly defined meanings” characteristic of Theory T tradition. Inany event, my own semantic pragmatism is more thorough-going than his, and morerobustly adaptive.67 It emphasizes the fact that good science commonly builds itselfup in layers of established reliability. Essay 1 explains why articulating a descriptivepractice in contextually sensitive patches often represents a wise linguistic policyfrom a data registration point of view; tyranny of scales considerations leave us noother choice. As language enlarges its applications in this manner, central words shifttheir registrational focus subtly through adjustments to local descriptive advantage. If these adaptive processes are not recognized, significant conceptual confusions mayensue, despite the fact that the terminological extension itself represents a great stepforward in science. Practical methods of addressing these disharmonies often requirethat different patches “talk to one another” in non-amalgamative ways, such as thehomogenization techniques of Essay 5. More generally, the processes of slowlyfashioning the patchwork policies of science into cooperative inferential harmony arefar different than simple Theory T presumption fancies. And that is the basis on

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Bernoulli principle

which my own anti-necessaritarian argument runs.Jim Woodward’s rich documentation of “cause”’s connections with the

counterfactuals of controlled manipulation do not cover every purpose for which weemploy the word within our practical employments. But that doesn’t indicate a faultin Woodward’s work; it merely demonstrates a simplistic semantic rigidity on the partof the critics who admonish him for not disclosing “cause”’s deepest groundings(“‘cause’ must possess a constant root meaning,” they contend). The analyses of“cause” they instead offer invariably strike me as either vapid or flatly wrong. Incontrast, many developmental episodes in the history of science can be greatlyilluminated by utilizing Woodward’s observations, because they track the ontogeny ofthe layered architectures in which science assembles its ingredients.68 Here’s acharacteristic illustration, involving the conceptual confusions that commonly attachto the familiar notion of “pressure.” Historically, our understanding of this notionbegins by considering the palpable resistance that a fluid displays in response to anyexterior attempt to compress its bounding surface in some way, e.g., with a piston. Inthis manner we establish water in bulk appears to observe a constant volumeconstraint:

No manipulation, including increased pressure on its boundaries, upon a flaskof water can cause any change in the volume of any packet of the fluid.

Note that this tenet directly reflects manipulations upon the boundaries of a containerof exactly the character that Woodward highlights. At this stage, “pressure” appliesonly to the degree of directed mechanical effort we can extract from the fluid througha mechanical arrangement, in the manner of a hydraulic press. It is not a priorievident that our fluid possesses any analog within its interior, away from walls onwhich it presses. However, our constant volume datum provides us with a causalcriterion for extending the applications of “pressure” inward from its boundaries, onthe grounds that if large blocks of fluid can maintain constant volumes correlated withtheir velocities and measured wall pressures, then smaller portions of the materialdeeper inside the container can be expected to do so as well. This applicationalextension allows us to speak meaningfully of an “internal pressure” varyingthroughout the fluid, away from the manipulations we can perform upon its boundingedges. FIG: BERNOULLI PRINCIPLE Here’s theexplanation I found on Wikipedia on the day that Ilooked up “pressure”:

Bernoulli's principle for steady state flow canbe derived directly from Newton's 2nd law. Ifa small volume of fluid is flowinghorizontally from a region of high pressure to a region of low pressure, then

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volume change required in a compressive wave

scalar versus directional pressure

there is more pressure behind than in front. This gives a net force on thevolume, accelerating it along the streamline... If a fluid is flowinghorizontally and along a section of a streamline, where the speed increases itcan only be because the fluid on that section has moved from a region ofhigher pressure to a region of lower pressure.

Note the central role of our constant volume constraint in this argumentation. Everyhigh school physics student is taught to reason in this fashion and this link betweenconserved volume and pressure allows us to talk of an unmanipulated “pressure”factor distributed unevenly throughout the interior of the fluid.

But this applicational extension readily generates a “paradox” that sometimesoccurs to budding young scientists:

Gee, the volumes that must remain constant in a fluid can be relatively largeand complicated and fluids aren’t very smart. But it’s very hard to distort anyobject on one side without temporarily affecting its volume. Thus considertwo blobs of fluid A and B situated next to one another. Waves travel throughwater at a finite rate, so when we first press on blob A, won’t it need to firstcontract for a short period and then reexpand, so that it transmit a heightenedforce to B when the two blobs again regain contact? In other words, watercan only maintain a simulacrum of our constant volume principle. Over shortintervals of time, the principle must fail.

Indeed, we can directly observe the time lag a fluid must suffer before it regains itsformer volume by considering the die swell within a tube of tooth paste: it takes ashort period of time before the paste fully “remembers” its uncompressed volumeinside the tube.

When we think in this vein, we have tacitly shifted to an alterative form of“cause” and “effect” investigation that ponders the processes that allow waves andallied disturbances to ripple throughinteriors of continuous media. As Essay2 notes, “cause” often redirects itsattention to such circumstances as well. We are naturally led to a picture of wavetransmission very much like we’d expectto see if we compress a train ofcontacting billiard balls from the left. FIG: VOLUME CHANGE REQUIRED Each successive ball must first contractand then reexpand to transmit acompression wave through their interiors.

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projecting a constant volume constraint onto a fluid flow

As a result, we require a revised notion of “internal pressure” that eschews itsconstant volume antecedents and allows a small packet of water to temporarily store acertain quantity of strain energy linked to its volumetric compression. And a furthersubtle alteration emerges as well. Normally, a pressure represents a scalar, non-directional quantity. FIG: SCALAR VERSUS DIRECTIONAL PRESSURE Whenwe compress a blob of fluid with a piston, it reacts by increasing its internal pressureequally in all directions. But wave motion requires a directional notion of pressure;our wave must remember the orientation of its original pushing. Indeed, as we beginto employ “pressure” in this further extended manner, we are well along the way tothe subtle concept of tensorial stress, first articulated by Cauchy, that further allowslittle blobs of matter to press on one another in oblique directions related to shear.69

Unfortunately, but inevitably, these varied employments of “pressure” oftennestle together promiscuously in modern descriptive practice. Thus, it is common(and deductively efficient) to treat normal water, in its mid-stream flow, as an“incompressible fluid.” Typically, we simply add a constant volume constraintequation to the standard Navier-Stokes formulas for our fluid. Mathematically, thissupplementary stipulation represents a projection of a higher level demand(symbolized by the winches of Lagrange multiplier corrections70) onto the lower levelplane of fluid blob behavior. FIG: PROJECTING A CONSTANT VOLUMECONSTRAINT By introducing this seemingly innocuous higher level constraint, wedeprive our equations of any capacity to determine what the “absolute pressure”inside our blobs is like, i.e., the stored strain energy responsible for wave motion. Inthe usual jargon of the physicists, the concept of “absolute pressure” becomesindeterminate in constant volumecircumstances.

Unsurprisingly, all of this provesutterly mystifying to budding scientists. Because pedagogy often recapitulatesontogeny, young engineers are taught theconstant volume employment of“pressure” first and only later warned thatthey must beware of confusing this non-absolutist notion of “head pressure” with acompressive “absolute pressure.” Everystudent of the subject at some pointexperiences vertigo with respect to thetrue “meaning” of “pressure,” illustratingmy theme that the asking price of an

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1. George Eliot, The Mill on the Floss (London: Penguin, 2003), p. ?

2. “The Handmaiden’s Tale,” Philosophical Studies 160 (2012), p. 3.

3. Theodore Sider, “Introduction” in T. Sider, J. Hawthorne and D. Zimmerman,eds., Contemporary Debates in Metaphysics (Hoboken: Wiley-Blackwell, 2007), p.??. I should observe that I have somewhat exaggerated the methodologicaldivergencies between Paul and Sider for my own narrative purposes.

effective reasoning scheme is often some form of conceptual headache. On the other hand, good science frequently builds up efficient descriptive

architectures in this layered manner, resting its extended applications upon platformsof well-established prior applications to which the new employments prove slightlyaskew. This is a strongly Duhemian observation, amplified in greater detail in Essay4. Engineers learn to master these adjustments through acquiring a tacit sensitivity tocontext through somewhat mystifying directive instructions such as “Oh, that’s only ahead pressure.” Rarely do they reflect on the semantic fact that the criteria for“pressure” vary considerably across applications, although a sharply articulated“paradox” can bring the underlying discrepancies to the fore. In reviewing this history, we see that the words “cause” and “effect” shift theirapplicational foci every bit as liberally as does the word “pressure”; indeed, the twoadjustments operate in coordinated parallel. Throughout these essays, I repetitivelystress the fact that it is overall healthy for a linguistic practice to assemble itsdescriptive policies in an adaptive fashion, rendering the prospects for an invariantmetaphysics of cause and effect dubious. One of the great merits of Woodward’sstudies is that he draws our attention to the centrality of manipulationistconsiderations within the histories of terms like “pressure.” We noted that analyticmetaphysicians complain that Woodward fails to isolate the core metaphysicalmeaning behind “cause,” on the grounds that we must have surely acquired suchinvariant notions within our juvenile intervals of early apriori inoculation. To makesuch presumptions is to misjudge the meandering currents that carry healthydevelopments forward within both science and language.

Endnotes:

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4. The algebra involved is especially easy when one starts with an impulse functionas I have, giving rise to the “fundamental solutions” of Essay 8. But if our startingdata is more complex, then the algebra involved in coordinating all our little fivepoint stencil requirements becomes rather daunting for plodding humans, althougheasy enough for a computer.

5. More exactly, critics of “cause” like Ernst Mach and Bertrand Russell claim thatphysics prefers to articulate its important relationships in “(differential) equations”rather than Humean talk. In these remarks, they patently fail to attend to JacquesHadamard’s vital distinctions amongst the “signatures” of various modelingequations (which align closely with the different forms of explanatory strategycanvassed in these essays). Not all differential equations are born equal (if Iunderstand him correctly, John Venn makes a similar observation in Empirical Logic(London: MacMillan, 1889), p. 68 and elsewhere). Writers such as Wesley Salmonand Philip Dowe regard Venn’s views as precursors of their own “causal process”accounts (see Dowe, “Causal Process Theories” in H. Beebee, C. Hitchcock and PMenzies, The Oxford Handbook of Causation (Oxford: Oxford University Press,2009). In my opinion, these authors deviate in unmotivated ways from themathematical underpinnings highlighted here, which were clearly articulated byHadamard in his classic Lectures on Cauchy's Problem in Linear Partial DifferentialEquations (New York: Dover, ND). I once asked Wes Salmon about Hadamard buthe was unfamiliar with the work, although it is now standard within appliedmathematics. Sheldon Smith supplies a good philosophical exposition based uponHadamard in “Resolving Russell's Anti-Realism About Causation," The Monist,83(2) 2000. My own position, of course, that this diagnosis only captures one of theseveral poles to which the word “cause” is naturally attracted.

6. The “little elements” of Essay 8 supply a case in point; WS supplies many otherexamples of this ilk.

7. Euler’s rule is described more fully in Essay 2. The diagram exploits theassumption that the arrow’s horizontal velocity will remain constant, allowing us tolocate our Euler’s rule time steps on a single chart.

8. To paraphrase the redoubtable George W. Plunkitt.

9. In truth, the slip-driven bow impulses within a real violin string do not smoothout into standing wave patterns as rapidly as conventional discussions assume, but

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continue to travel back and forth across the string in more localized pulses. We’lllater observe that a standard Fourier analysis of musical tone in terms of sine wavesapplies more aptly to the “acoustic far field” generated within the interveninginstrument body and the intervening air (various psychological aspects of tonalperception enter the picture as well). We shall return later to the question ofwhether the sine wave modes represent “true properties” of the string or not.

We might also observe that a relaxation into standing wave pattern representsa form of “forgetfulness” on the part of the string akin to the loss of control thatarises when a coherent pressure is converted to incoherent heat. In addition, someof our perception of “pure tone” is psychological in origin as well.

10. Many of my complaints about the shallowness of Theory T analysis in Essays 2and 8 directly relate to this observation.

11. See Essay 9's discussion of Sturm-Liouville problems.

12. Our string’s adjustments in gravitational potential energy are so negligible incomparison and are generally ignored.

13. That is, we now consider an interior modeling in which the “k” in our waveequation is no longer a constant but varies with x: 2y/t2 = k(x) 2y/x2.

14. Observe that in seeking these eigenfunction decompositions, we are relocatingour string problem within a different “explanatory landscape” than we utilizedpreviously when we conceptualized its circumstances in straightforward marchingmethod terms (which represents a computational setting we’d normally like to avoiddue to its greater propensity for falling into serious numerical error).

15. Mathematically, our attention shifts from our original hyperbolic wave equationto an elliptic reduced wave equation, after we factor away time through separationof variables. The analogies to the “looking for equilibrium” strategies of Essay 2are patent.

16. In this instance, our cave man’s reasoning transfer probably operates in adirection opposite to that we discussed, for, most likely, he was not skilled in graphpaper plotting. Opportunities for reasoning transfer often operate as two waystreets.

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17. Sometimes the “clock of time” we employ in these calculations conveys nophysical significance whatsoever–see appendix 2 to Essay 2.

18. Making Things Happen (Oxford: Oxford University Press, 2003). Woodwardlargely confines his attention to how “cause” applies within straightforwardexperimental circumstances but we’ll find that close cousins to his characteristicmanipulationist conditionals arise within a wider set of circumstances whereconsiderations of how one variable controls another arise. One then seeks areasoning architecture that can mimic these relationships. We frequently employcausal locutions to articulate the key dependencies needed within these newlinguistic arrangements.

19. In more formal terms, we have shifted from a traveling wave decomposition ofour string’s behavior to a standing wave analysis, which provides a great gain inreliability in suitable circumstances. It is sometimes claimed that many of thegreatest advances in nineteenth century physics were accomplished through Fourier-like adjustments of the sort canvassed here.

20. Critics of “cause”s place in physics, such as Ernst Mach, often appeal to these“anthropomorphic” ingredients as evidence that the notion merely represents a relicof animistic superstition. To which we should rely: (1) control questions representa significant aspect of science as well but (2) our everyday tools of languagemanagement allow us to filter away these ingredients when needed.

21. For the mathematical underpinnings of this notion of lifting, see Essay 4.

22. In a “virtual” manner–see Essay 7 for an explanation of this, as well as a fullerdiscussion of the possibility spaces relevant to virtual work considerations.

23. As noted in Appendix 1 to Essay 1, the philosopher Amy Thomasson declaresthat applying causal talk across conceptual protectorates is a mistake. I don’t seewhy; normal usage doesn’t comport with such a restriction.

24. As Essay 1 explains, reasoning within a single level framework enjoys certainadvantages over multiscalar schemes but their descriptive asking price oftendemands that we employ quite fancy mathematical constructions in modeling“simple” physical arrangements such as cracks or entrapped stresses. For more onthe contrast between single-level and multiscalar approaches to modeling, see theconcluding section of Essay 4.

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25. With the rock perlite, a tenfold volume change can suddenly appear.

26. Often these conditionals report on how single variable adjustments ormanipulations at one scale level will induce corresponding effects upon other scales. These conditionals must usually fit together as large but coherently focusedcollections, for these supply the basic data underlying the formal search spaces weemploy in reasoning by successive approximation, as when we seek the overallstatic balance of our linked seesaws by tweaking our children in a quasi-experimental fashion. Essay 7 explicates these requirements in more mathematicalterms.

27. Although I have put these “psychological” characterizations in quotes, this is theway that materials scientists actually talk about springs–see Essay 3 for more onthis.

28. It should be observed–the rubber band example of Essay 4 provides a case inpoint-–that such distinctions multiply as the higher level structures in a solid becomemore complex. We can intuitively “heat up” a rubber band without stretching it,through increasing the wigglings of the polymer chains that comprise the material. But these linkages retain a certain coherence amongst themselves that can berecovered through suitably delicate interventions. Sometimes tiny creatures canexploit the residual forms of coherence encountered at their scale level, which areforever lost to ham-handed giants such as ourselves.

29. As the appendix to Essay 1 explains, I believe that the word “common sense”itself connotes a basic mastery of a wide range of potentially useful reasoningstratagems, to be mixed and matched as applicational circumstances demand.

30. Philosophical Investigations (Oxford: Blackwell’s, 2001), §67.

31. Sometimes analytic metaphysicians claim to be engaged in some Saul Kripke-like investigation of “a posteriori necessities” but Eli Hirsch’s assessment strikes meas just:

I think it is clear that many of the most prominent arguments in recentontology (in Chisholm, Cartwright, Thomson, Lewis, Shoemaker, Unger,Sosa, Van Inwagen, Van Cleve, Sider and many others) are a priori in therelevant sense. Insofar as some (not all) of these philosophers will now andthen gesture towards the fact of empirical science, ...I think ...that [theymerely] adopt the speculative tone of high-level theorists rather than the

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tone of philosophers engaged in straightforward conceptual or linguisticanalysis. That may well be, but their main arguments, whatever theirspeculative or theoretical tone, are a priori rather than empirical.

(“Ontology and Alternative Languages” in David Chalmers, David Manley andRyan Wasserman, eds. Metametaphysics (Oxford: Oxford University Press, 2009),p. 233).

32. Quoted in Roberto Bonolo, Non-Euclidean Geometry (New York: Dover,2010), p. 25. CHECK

33. I am thinking less of non-Euclidean geometry per se and more of the generalconcept of manifold, assembled by stitching together local sheets of computationalpolicy. Riemann’s innovations on this score are inspired by earlier work on geodesysuch as Gauss’, through considering the variety of maps required to cover the earthin an appropriate fashion. These developments have played a large role in my ownthinking as to how judicious “meanings” should be assembled, as Essay 9 makesclear.

34. Indeed, the odd behaviors of analytic functions of a complex variable (such assquare root or the exponential) provides an excellent model for how early aprioricertitudes get undermined under linguistic extension. Pioneering mathematicianssuch as Euler initially mapped out the basic behaviors of familiar functions over thecomplex numbers through multiplying their factors naively. Eventually they realizedthat these same computational rules break down if the numbers multiplied lie too farapart on the complex plane. Indeed, the reasoning tools they had trusted in theirinitial explorations remain valid only if their operations transpire upon a single sheetof the multi-valued Riemann surfaces upon which analytic functions naturally live. In the mathematician’s jargon, these multiplication rules are valid locally but notglobally. Such global discrepancies are often masked by the psychological fact(discussed in Essay 1) that we generally conceptualize applicational circumstanceswithin restricted scenarios linked to scale size. Through this masking effect, we failto notice that our local, early apriori assumptions about “part” and “whole” (whichrepresent the reasoning rules that mereology attempts to capture) becomeproblematic under scale shifts. Similar factors obscure the fact that classicalmechanical practice does not happily submit to any univalent axiomatization. SeeWS, pp. 193-203 for more or this.

35. And control variable questions which are very much like them.

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36. Nelson Goodman, Fact, Fiction and Forecast (Cambridge: Harvard UniversityPress, 1983). David Lewis, Counterfactuals (Cambridge: Harvard University Press,1973). Robert Stalnaker, "A Theory of Conditionals" in Nicholas Rescher, ed.,Studies in Logical Theory (Oxford: Blackwell, 1968).

37. They confidently hold to this opinion, despite the fact that few of them couldriffle through the pages of a standard physics primer and reliably locate the “laws”therein (I find that such authors frequently confuse laws proper with the differentialequations one employs within a concrete modeling, despite the fact that the latterpatently rely upon factors that are not “law-like”). In truth, the doctrines we call“laws” in real life comprise a rag-tag collection of classifications tied together onlythrough various forms of historical accident. I find it astonishing that this fuzzynotion is now viewed as a key ingredient within our early apriori arsenal (in contrastto “laws of causality” of earlier times). Have our metaphysicians ever attempted toteach the notion of “law” to a seven year old? On the other hand, construing “laws”as the constitutive recipe instructions required to build up the hyperbolic operatorsof an evolutionary modeling strikes me as a fairly firm notion and close to theintuitive notion to which philosophers often appeal. In present circumstances,however, our analytic metaphysicians are employing “law” in considerably moreelastic manner.

38. In Essay 7, we find that Woodwardian facts are commonly collated intorestricted possibility spaces within applied mathematics. The formal manner inwhich such spaces and their associated norms assist our projects of physicsavoidance are explained more fully there.

39. This is the chief origin of the “missing physics” I describe in "Determinism: TheMystery of the Missing Physics," British Journal for the Philosophy of Science2008. Specifically, Newton’s second law, construed as fi = ma, demands a listingof every special force law fi cited in this summation. But classical tradition offersno canonical candidates for these fi beyond universal gravitation. Modelingpractice generally sidesteps these lacunae through appeals to higher scale rigiditiesand similar tactics.

40. New York: Dover, 1957, retitled as Fads and Fallacies in the Name of Science.This delightful book is highly recommended to all would-be philosophers.

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41. This complaint about contemporary philosophical thinking runs persistentlythrough this entire collection of essays, but is especially highlighted in Essay 2.

42. In Essay 8, I complain about how philosophers frequently abuse this usefulphrase.

43. Several authors commendably render this formal identification with evolutionarypractices more explicit, particularly L.A. Paul and Ned Hall, Causation: A User'sGuide (Oxford: Oxford University Press, 2013) and Tim Maudlin, The MetaphysicsWithin Physics (Oxford: Oxford University Press, 2007), Chapter 1. We shouldobserve, however, that such authors employ “initial conditions” in a wider thanoriginally intended sense (it should be restricted to delineations of a target system’scurrent phase along a Cauchy surface) and incorporate many issues of systemconstruction that, properly speaking, characterize how the conglomeration holdstogether over extended stretches of time. As other essays make clear, persistentphilosophical abuses of “initial” and “boundary condition” will curl the hair of anydevoted reader of Jacques Hadamard!

44. As Essays 4 and 7 illustrate, the counterfactuals reached through directexperiment often need to be tweaked into their virtual displacement cousins toprovide the requiste guidance. Pierre Duhem, The Origins of Statics (Dordrecht:Kluwer, 1991) is the classic reference on these topics; Danilo Capecchi, History ofVirtual Work Laws (Berlin: Springer, 2012) is a more recent survey.

45. Philosophers often distinguish between “epistemic” and “doxastic” possibilities,but the “instructions for reasoning” aspects of our specialized possibility spacesindicate that that division is not as clearly marked as these authors presume. Essay7 warns of the dangers of inflating smallish, focused notions of “possibility” intopossible world behemoths.

46. I found this quote in H. J. S. Smith, “On Some of the Methods at Present in Usein Pure Geometry” in Collected Mathematical Papers Vol. 1 (Providence: AMSChelsea, 1965) p. 6). He credits the remark to an earlier British mathematician,Richard Woodhouse.

47. And, I submit, improving human thought generally. See the preface and WS formore on this theme.

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48. In his Papers in Metaphysics and Epistemology (Cambridge: CambridgeUniversity Press, 1999).

49. Early battles over “functionalism,” “supervenience,” “grounding” and so forthfocused upon the question whether a syntactic expression that contains anabundance of quantifiers (especially of a functional stripe) elevates the trait beyondthe allowable orbits of “physical kinds” and into the nether spheres of psychology orthe other special sciences. Hilary Putnam started much of this, presumably inspiredby his work in recursion theory (where layered stacks of quantifers genuinelycorrespond to important hierarchies of mathematical behavior). In my antique“Honorable Intensions” (in Wagner and Warner, eds, Naturalism: A CriticalAppraisal (South Bend” Notre Dame Press, 1993)), I argue that this entire project ofsorting out “kinds” through syntax is misguided within a physical context.

50. Old theorems due to Joseph Liouville and others establish this transcendence;see J.F. Ritt, Integration in Finite Terms: Liouville's Theory of Elementary Methods(New York: Columbia, 1948). The essential role that set theory plays in resolvingthese problems is discussed in Essay 9 under the heading of Sturm-Liouvilleproblems.

51. Example from Essay 5: the isotropic scaling behaviors of well-made steel failwhen we reach the level of its component grain, yet we relentlessly plow past theselimitations in setting up the standard modeling equations for our subject.

52. As mentioned before, these observations with respect to “relaxation times” applyto many vibrational systems but not, as it happens, completely to violin strings. Rather than switching examples, I’ll stick with violin strings and engage in a bit oftemporary pretense, which we’ll lift later in the essay.

53. In W.V. Quine, From a Logical Point of View (New York: Harper’s, 1953).

54. Many scientists operate from a position that might be called particle physicsmyopia, as if the asymptotics of controlled scattering experiments is the only issueof descriptive interest within science. That, assuredly, represents a very centralscientific concern but we also need to consider how such issues fit together with therest of nature, especially with respect to the mathematics employed. Note that Siderapparently commits his future T to also providing “cosmologies” in the cut freemanner discussed in Essay 4; this strikes me as another formal issue upon whichphilosophers should not legislate prematurely. I also detect the secret hand of ODE

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expectation continuing to play a shaping role here, but this thesis is harder toestablish.

55. More properly, a generalized Sturm-Liouville factorization into eigenfunctions;the phrase “Fourier analysis” sometimes restricts its attention to sine wavedecompositions, which may not represent eigenfunctions of the target system. Therole that set theory plays in legitimating these traits is discussed further in Essay 9.

56. Mathematicians say that our problem “admits separation of variables”–it can befactored into separately addressable questions.

57. I am often surprised that logic-influenced philosophers of science often write of“definitions” in a loose manner that fails to accord with the strict standards of“extension by definitions” developed by the logicians. I often cannot follow thecontours of their argumentation as a result.

58. The great paradigm for this kind of treatment can be found in L. Prandt’sintermediate asymptotics approach to boundary layer theory. See G.I. Barenblatt,Scaling, Self-similarity, and Intermediate Asymptotics (Cambridge: CambridgeUniversity Press, 1996). In a passing note in WS, I opine that the strategy behindvague predicates such as “baldness” should be investigated from this point of view. I still maintain that opinion, but have never followed up on the hunch.

59. Galileo, “Selections from ‘The Assayer’” in Maurice A. Finocchiaro, ed, TheEssential Galileo (Indianapolis: Hackett, 2008).

60. In truth, aspects of both themes appear in both Paul and Sider, but I havefavored a crisper allocation for dialogical reasons.

61. His concerns are further discussed in Essay 9.

62. Such isolationist proclivities are further illustrated in Essay 8, in application toDavid Armstrong’s methodological claims.

63. The Lion with Gerald Clark and His Caribbean Serenaders, “Believers in theLand of Glory,” Decca 17307.

64. Oliver Wendell Holmes, Jr., “The Path of the Law,” 10 Harvard Law Review457 (1897).

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65. “Two Dogmas of Empiricism” in From a Logical Point of View, op. cit.

66. In Constructing the World (Oxford: Oxford University Press, 2014), DavidChalmers dismisses Quine’s and my own reservations about linguistic change inmore or less these terms, albeit expressed in a considerably different manner. Foran equally brisk reply, see my “David Chalmers Versus the Boll Weevil,”Philosophy and Phenomenological Research, 89(1) 2014.

67. Caveat: traditional pragmatists are often deflationists about meaning andreference, which I am surely not. I merely think that referential ties to nature worldarise ultimately from language’s practical entanglements with it, which commonlyinvolve rather complex forms of strategic information registration.

68. Essay 4 presents another striking illustration, in which Woodward’s emphasescomport nicely with Duhem’s structural analysis.

69. Since fluids don’t sustain any shear, we reach an intermediate conception that issometimes called “absolute pressure.” An excellent survey is Clifford Truesdell,“The Creation and Unfolding of the Concept of Stress” in Essays in the History ofMechanics (Berlin” Springer-Verlag, 1968). These directional subtleties havedirectly affected philosophy as well; Kant’s program in his MetaphysicalFoundations of Natural Science (Cambridge: Cambridge University Press, 2004)runs in difficulties of exactly this nature. It is worth noting that a scalar pressuredegrades the coherence of applied directional work in a more extreme manner than adirectional pressure, without converting it to totally incoherent heat. The subtletiesof energy degradation will reappear in many of these essays and the maintenanceand loss of coherent control is an important theme in Woodward’s work.

70. The appendix to Essay 7 explains more fully what I have in mind here. Itdiscusses how the word “force” behaves under impositions of constraint in a mannerthat closely resembles the present discussion. However, constant volumeconstraints are weaker than the rigidity constraints discussed in Essay 7 andaccordingly induce subtler shifts in representational significance.