our task

Essay 9

A Second Pilgrim’s Progress

As I walked through the wilderness of this world, I lighted on a certain placewhere was a Den and I laid me down in that place to sleep: and, as I slept, Idreamed a dream. I dreamed, and behold, I saw a woman clothed with rags,standing in a certain place, with two books in her hand and a great burdenupon her back. I looked and saw her open those books and read therein;and, as she read, she wept and trembled; and, not being able longer tocontain, she broke out with a lamentable cry, saying, "What shall I do?" --John Bunyan (with apologies)1

(i)

The distressed pilgrim I witnessed was Penelope Maddy and the twofearsome books she held were Quine’s From a Logical Point of View andBenacerraf and Putnam’s Philosophy of Mathematics, second edition.2 She wasembarking on a journey that she reported upon in her SecondPhilosophy of 2007. FIG: OUR TASK Maddy and I arewayfarers of a common cloth; we share the conviction that thetwo influential texts indicated have guided philosophy ofmathematics in unfortunate directions over the past severaldecades, leading along pathways that seem beguiling at first butwhich eventually entrap the traveler in dank sloughs andunwholesome dens of ignominy. This misdirection transpiresunder the alleged imperatives of “naturalist philosophy,” whichprima facie sounds as if it should represent a Very Good Thingbut somehow leads its adherents to mutilate mathematics inridiculous ways.Second Philosophy can be read--or, at least, so I’ve read it--as a valiant attempt toreorient the compass of naturalism along a more reasonable axis. She begins herexplorations by vowing that she will not be distracted from her mission by the usualcavils of philosophical skeptics and will strive to approach all issues in thecustomary vein of an empirical scientist.3 Nor will she accept methodological claims

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reformers

(e.g., Quine’s “mathematics is posited for the sake of physical science”), for whichlittle evidence can be adduced in practice.

Maddy is concerned that adherence to such constrictive percepts is apt to clipthe wings of mathematics’ most stirring developments, for Quine’s characteristicemphasis upon the low cravings and brute advantages of physical expediency seemaltogether alien to the ever-widening investigative spirit that animates most modernmathematics. In that spirit, he regards “the more gratuitous flights of higher theory”as mere “mathematical recreation...without ontological rights.”4 She stresses thatthe dominating modes of development within real life set theoretic endeavor appearsto be ones of maximizing the range of structures of potential interest, in sharpcontrast to Quine’s crabbed gospel of “minimize and Ockhamize.” All of theseobservations strike me as entirely on target.

She is further troubled by a band of zealots whohave set out, armed with pitchforks, to reform theexcesses of mathematical postulation under the banner of“naturalism.” FIG: REFORMERS Surely, somethinghas gone awry when serious authors fancy that they assistthe Progress of Man by rewriting regular mathematicalassertions within weird codes or by hiding such assertionsbehind sentential operators that signify “I don’t believe itbut I want to use it.”5 Later in the essay we’ll consider the philosophical joy juicethat inspires these strange enthusiasms.

Nonetheless, Maddy’s own travels convey her to a final destination that I finduncomfortable, to an unfettered republic of Pure Mathematics whose governingtenets are entirely determined by the “practices” of its democratically nominatedinhabitants--the mathematicians. Like most would-be empiricists, I’d rather notallow any portion of language to escape so flagrantly from the tribunal ofexperience. It is a sad but undeniable fact that all of our practices, without adequateresponsibility to exterior corrective, can fall into the stagnating grasp of cults and/ora priorist philosophy. Even a classificatory predicate of minimal utility such as“contains orgone” can enjoy a lengthy and fulsome linguistic life if its employmentremains sustained within the practices laid down by some mutually reinforcingcombination of gurus and disciples, supplying, perhaps, a psychologically fulfillingform of life but not one that displays an improving empirical grip on nature.6 WhenWittgenstein complains of set theory:

In a dark cellar roots grow yards long,7

he expresses the concern that even earnest mathematical labors can fall into a trap of

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improperly constrained development. As it happens, I will argue that it is wrong tolook upon set theoretic practice in this deprecatory manner. Nonetheless,Wittgenstein’s worries capture, in general terms, the snares of unproductivedisengagement against which we should remain ever vigilant, within every field ofendeavor, including mathematics. So I like not Maddy’s baseline appeals to the raw“practices” of the subject.

However, her own pilgrimage neglects an alternative pathway that meritsfuller exploration, for it leads to a softer form of naturalistic resolution.8 This trailbegins by objecting to the pretense of presuming that scientists “posit” all of theirscientific and mathematical doctrines in one glorious fell swoop, a view that I call a“man proposes; nature disposes” portrait of theoretical endeavor. Quine,Benacerraf and their many followers regard this assumption as a harmlessidealization that allows for a simpler discussion of philosophical essentials, but Ibelieve that such an artificial condensation into tidy posits obscures many of the keyfeatures of science’s ongoing entanglements with mathematics, a troublesomerelationship that we might roughly characterize as “mathematics’ work is neverdone.”

Throughout these essays, I have stressed the fact that nature rarely arrangesits affairs for our calculational convenience but forever forces us into seeking cleverwork-arounds for improving our computational lot in life (this is what the phrase“physics avoidance” signifies). The natural history of man as a biological speciesrequires that we attend to our ongoing struggles in practical improvement, just asregular natural history inquires about the capacities of frogs and mayflies in an alliedecological vein. In doing so we must study, at closer quarters than idealized “manposits mathematics and science” assumptions encourage, the detailed manners inwhich mathematical thinking promotes these ongoing improvements. This projectrequires that we focus more intently upon the detailed character of effectivestrategies, how we recognize their descriptive merits and how we fashion linguisticpolicies that can take further advantage of the opportunities offered. Throughout allof these improving activities, much wider swatches of mathematical thought playcrucial roles, far beyond the narrow horizons anticipated within the writings ofQuine and Benacceraf. If philosophers frame a conception of naturalistic obligationthat does not reflect these developmental issues squarely, improper presumptionswill have surreptitiously compromised their “naturalism.”

And this is exactly what seems to have gone amiss with the pseudo-scientificquests that animate many of today’s philosophers. Through their seeminglyinnocuous appeals to Theory T posits, the real life struggles we confront in dealing

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beware of sleeping on enchanted ground

with a largely uncooperative natural world are buried and neglected, and, with them,the strongest empiricist rationales for developing higher mathematics. Accordingly,if we can liberate the practicalities ofstrategic improvement from their TheoryT imprisonment, our pilgrimage towardsa more balanced appraisal ofmathematical endeavor will fare better,having cast aside the baleful influenceof the two fearsome books with whichwe began.

None of the milder naturalism weseek contends in a Quinean frame that“mathematics is entirely posited forscience’s purposes.” Indeed, I doubtthat either science or mathematics enjoys any clearly identifiable purpose except toarticulate “things that seem correct.”9 But a greater attention to ecological detailshows that the various reductive projects of which Maddy complains plainly restupon severe misunderstandings of the roles that broad avenues of mathematicalthinking play in gradually improving our computational place in nature.

With respect to inspiring books, I would have never thought to articulate myruminations in this essay’s manner had I not first read Maddy’s Second Philosophy. We are pilgrims of a similar faith and I want to acknowledge those affinities here. However, I must warn readers that our own journey may prove occasionallywearisome, and mired in detail, as we pass through philosophical analogs of VanityFair and the Valley of the Shadow of Death. This is because the chief obstructionswe shall face represent imps of Over-Schematization--the deceptive tales aboutscientific methodology that flatter our conceits that we can ably understand howapplied mathematics operates without much consideration of the twists and turns ofreal life conceptual development. That great compendium of methodologicalfolklore that I characterize as the Theory T syndrome plays a large role inencouraging this enchanted self-assurance. FIG: BEWARE OF SLEEPING ONENCHANTED GROUND As a corrective, we need to look at a lot of Little Thingsin closer focus. The road to perdition is paved with inadequately examinedexamples.

(ii)

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a mayfly’s strategic opportunity

So this is where our own pilgrimage begins: what do we currently know aboutour computational position within nature? How ably can we expect to employreasoning algorithms effectively against the backdrop of a complex natural setting? And through what intellectual means should we expect to establish those policiesreliably and to improve their operations further?

So employed, the phrase “mankind’s computational place in nature”represents a simple specialization of the general notion of an environmentalopportunity in the sense that biological organisms find themselves confronted with arange of exploitable options, according to the physical conditions prevailing at thesize and time scales upon which they live. For example, little inlets along theshoreline of a stream can act as frequency traps, in that they orchestrate their captivewaters into standing wave patterns in much the manner that a master of the musicaljug sets the air within his vessel into pleasingcongruities. The slow moving coherence of theentrapped water offers the enterprising mayfly anopportunity to perch on the side of the cavity andreap delightful delicacies at its leisure.10 FIG: AMAYFLY’S STRATEGIC OPPORTUNITY

The circulating motions within a creeksidepocket provide our insect an opportunity forfeeding not available to bulky objects such asourselves (even if we wish to eat flotsam). Many marvelous studies on scale in thebioengineering literature have made it clear that the characteristic “worlds” ofhumans and sand flies, fish, paramecia, etc. differ greatly from one another.

[T]here are inescapable biological consequences of size and design. Forexample, swimming with the aid of cilia or flagella is possible only for verysmall organisms, and fishes use a different propulsive mechanism. Aparamecium covered with cilia swims many times its body length in asecond, but a giant shark covered with cilia would get nowhere. The laws offluid mechanics can, in a more formal way, explain why microorganismsand fish, from this point of view, seem to live in different worlds.11

These differing worlds (= different varieties of environmental niche) are generallydetermined by different ranges of dominant objects and properties, in the sense ofrepresenting the characteristic manners in which an organism attends to the ambientdata available to it.

Consider, for later purposes, another, somewhat fanciful, illustration of these

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a frog’s strategic opportunity

a truant officer’s strategicopportunity

biological ideas. Certain frogs might wait until their flying prey execute tight loopsbefore they attempt to catch them. FIG: A FROG’S STRATEGIC OPPORTUNITY

Why might such a policy prove advantageous for ouramphibian friend, rather than simply attempting to impalethe fly anywhere along its path? For reasons we’lldiscuss later, the fly’s outward flight path can often becharacterized with fewer descriptive parameters than aregenerally required around the smoother parts its journey. This parametric opportunity allows the frog to frame abetter estimate of where the fly is likely to travel a fewseconds hence, without excessive demands upon itslimited capacities for observation and reckoning.

Such animals scarcely engage in any significant amount of computation(although our frog undoubtedly employs some analogical surrogate for analgorithm), but we do, on an increasingly overt basis as our scientific understandingof the world improves and becomes explicitly mathematical. In this respect, natureoffers us special computational opportunities where, through well-matched mixturesof observation and calculation, we can accomplish chores adequate to our purposes. Suppose we are truant officers eager to catch Jack and Jill in their hooky fromschool. FIG: A TRUANT OFFICER’S STRATEGICOPPORTUNITY How can we catch them? Trying toaugur their trajectories in most locales involves difficultcalculations offering unreliable results. However, theyare currently frolicking on a steep hillside from whichthey are likely to tumble, so we should calculate wherethe loci of lowest gravitational potential energy lie, andlurk there, waiting to pounce. Here we assume that,over the relevant stretch of the childrens’ history,gravitational attraction will prove the dominant factoraffecting their behaviors, operating in league withfriction to bring them eventually to a temporaryequilibrium. But whether this particular form of physicsavoidance strategy will work or not depends on our owncomputational swiftness and how long the children remain stunned. If the littlerascals get up too quickly and rush away, our clever gambit will prove for naught.

I’ve concocted this little parable to illustrate the general character of thephysical circumstances required to underwrite the success of the explanatory

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an uncomputable natural process

strategies canvassed in Essay 2 (specifically, the advantages of an equilibriummodeling over a straightforward evolutionary plotting). Thus a computationalopportunity simply represents the physical circumstances that render certain flavorsof reasoning policy effective.

One of our greatest adaptive advantages over the frog and mayfly traces tothe fact that we needn’t rely so heavily upon the sluggish corrections of naturalselection to uncover and exploit the environmental opportunities available to us. We can figure out our own strategies for catching errant children, thank you, Mr.Darwin. But how do we manage to cobble together these routines in this first place? And how do we adjudicate their reliability later on? Why do the multiscalarroutines of Essay 5 often supply better results than purist bottom-up or top-downestimations? For good and ill, our natural histories are shaped by the answers weprovide to such questions. As naturalists-in-the-making, philosophers should weighthese shaping considerations in the same vein that investigators within the branch ofbiology called biomechanics evaluate frog behavior against a backdrop of theforaging opportunities available on the scales of size and time appropriate to anamphibian.

Part of this appraisal must begin with a balanced assessment of the inferentialtasks that we cannot expect accomplish within nature’s confines. Like it or not,there are some outcomes that we’ll never be able to predict through calculation, nomatter how hard we try. To paraphrase a noted authority on the limits of ambition(Mick Jagger): “you can’t always calculate what you want.” Despite the giddyassurances of Kantians and like-minded philosophical schools, we possess no apriori assurance that nature is inclined to trim its behaviors in order to suit the feeblecomputational cloth in which we attempt to dress it.

Nonetheless, learning that we can’t calculate our way out of every naturalexigency needn’t prove particularly depressing. We can design a frictionless pinball

machine with bumpers labeled “0" and “1" where thecareening pinball, under most initial conditions, willprogressively bump out a time series of highly non-recursive 0s and 1s. FIG: AN UNCOMPUTABLENATURAL PROCESS Through brute calculation, wehave no way of replicating this sequence. So what? Itseems sufficient that we can understand the physicalprocesses involved, even if we can’t grind out thetrajectory with the accuracy required. Assigning ourconcrete computational capacities a fairly low grade

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Descartes’ constriction problem

doesn’t indicate that our general understanding of nature isn’t adequatelysatisfactory.

(iii)

Let’s begin our pilgrimage with a descriptive difficulty that Descartesconfronted. He worried that we possess insufficient mathematical capacities fortracking the adjusting geometry of a continuous flow, such as water flowing througha pipe constriction. FIG: DESCARTES’ CONSTRICTION PROBLEM Nominally, these problems stem from the fact that he also maintained that matter isessentially granular--each component particle (normally) possesses a fixed size andshape at each moment in time. And he further rejected the notion that any emptyspace could ever appear betwixt between these individual corpuscles–they mustalways remain in full contactwith one another–, so at everyparticular instant every spatialarray of fluid decomposes into atight mosaic of contactingparticles. But these particulatetiling patterns must alter as amoving fluid passes into anarrowed portion of a pipe; howare these rearrangementspossible? As I understand hisscientific doctrines,12 they provide clever mechanisms for splitting integral particlesinto smaller pieces and for rejoining them into different geometries later on. Theconceptual difficulties of our pipe problem stem from the fact that continuouslyadjusting processes of fracture and fusing must occur instantaneously at a dust-like(= infinitely small) level if no vacuum gaps are to appear as the fluid passes throughthe constriction. Descartes’ surprising “resolution” to this difficulty claims that insuch circumstances the feeble human mind, and its attendant reasoning tools, aresimply incapable of following such an infinitary process through all of its componentstages. We must bluntly acknowledge that natural processes frequently passthrough stages of “indefiniteness” that our finite minds cannot track with the limitedreasoning tools available to us. His disciple Jacques Rohault writes:

[Aristotle's followers] did not consider that equality and inequality areproperties of finite things, which can be comprehended and compared by

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human understanding, but they cannot be applied to indefinite quantitieswhich human understanding cannot comprehend or compare together,anymore than it can a body with a superflies, or a superflies with a line.13

And:[T]here are in nature things which are vastly more fine and subtle; we shallclearly see that what exceeds our imagination is not therefore impossibleand that it is not for us to presume, as many do, to set bounds to the powerof God.

Since God is not a deceiver, we are still assured that, whatever the hell exactlytranspires when water moves past a constriction, it will somehow obey the sameconservation laws that we can ably track geometrically through milder forms ofparticle fracture and reassembly. This divine assurance allows us to confidentlypredict that, after our moving mass of water regains a finite particle geometrydownstream in the pipe, its reconstituted particles will reemerge with exactly thesame total volume as before (albeit shaped differently) and, barring any intrusion offriction, will collectively carry the same “quantity of motion” as before (i.e., linearmomentum in this context). Relying upon these before-and-after reasoningassurances, we can ratify conclusions like Bernoulli’s principle a priori, despite ourinabilities to inferentially track the intermediate stages that link the incoming flow tothe outgoing flow across the two sides of our pipe constriction. In modern jargon,we have avoided the hard-to-describe “singularities” located within the pipejunction by asymptotically gluing together descriptive materials that capture thesmooth flow away from the problematic constriction. This intervening gluing ismonitored by various “connection formulas”–momentum conservation in thiscase–that connect up the divided pieces adequately without delving into their“indefinite” details.14 To this very day, scattering phenomena in physics arefrequently addressed in this tactical manner, a circumstance to which philosophersof science should closely attend, for the prototypical computational successes ofquantum mechanics rely heavily upon asymptotic matching of this general character. This topic will reemerge in our study soon.

Descartes’ guarded perspective represents an early articulation of a theme Ishall dub mathematical opportunism: nature offers only restricted occasionswherein we can fully follow her developing processes with the reasoning toolsavailable to us within mathematics. Such cavils, as I stress in Essay 5, needn’trepresent any sort of “anti-realism” with respect to the external world itself--Descartes was certainly a robust realist–, but merely reflect our limited capacitiesfor tracking generic physical processes accurately over long spans of time utilizing

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our “mildly transcendental” relationship to our goose’s flight

mathematical tools. The mere fact that natural processes may evade convenientcomputational footholds does not automatically render their workings inscrutable insome more mystical or disheartening manner. We have noted that there are lots ofsimple processes that we can understand well enough--our frictionless pinballmachine--but where we recognize that we may never be able to grind out anaccurate enumeration of their detailed events as they unfold over time.

(iv)

But what might a suitable “understanding of computational landscape” looklike? Somewhere--I can’t remember where!–, I have seen it remarked thatDescartes falls into his pipe constriction difficulties largely because he lacks thedifferential equation tools that can ably govern the flow of a fluid while it movescontinuously through a narrowed opening without forming vacuum-style gaps. Thisremark is true enough insofar as it goes, but it neglects some subtle considerationsto which we’ll later return. Before doing so, let’s explore the form of mathematicaloptimism upon which this remark depends, along with the standard hopes of mostTheory T postulationalists. Doing so requires a fair expanse of exposition.

Let’s start with a simple situation: a goose that flies over a flat plane subjectto lateral winds of a specified but varying strength. FIG: OUR MILDLYTRANSCENDENTALRELATIONSHIP Suppose that wepossess a pretty good differentialequation modeling of the factors thataffect our bird’s flight path along withappropriate initial conditions.15 Let’ssimultaneously bear in mind LordKelvin’s hard-headed admonition toattend to the “numerical reckoning” (=algorithmic computation) that we canactually carry out in suchcircumstances:

In physical science the firstessential step in the direction oflearning any subject is to find principles of numerical reckoning andpracticable methods for measuring some quality connected with it. I oftensay that when you can measure what you are speaking about, and express it

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in numbers, you know something about it; but when you cannot measure it,when you cannot express it in numbers, your knowledge is of a meager andunsatisfactory kind; it may be the beginning of knowledge, but you havescarcely in your thoughts advanced to the state of Science, whatever thematter may be.16

At this point let us recall that differential equations, considered in their ownrights, describe unfolding events only at an infinitesimal level, leaving us the stickyproblem of converting this data into usable data pitched at a finite size scale. Andthis finitary extraction task can prove enormously difficult, even for very simpleequations (differential equations are apt to sit there and not say nuthin’, no matterhow deftly you may reason on them). Often the best we can automatically hope foris that we will be able to squeeze in upon the target system’s evolving behaviorsthrough some contracting sequence of numerical approximations (or seriestruncations) of the sort I will now illustrate.17 To be sure, there are clevererinferential tactics we can sometimes employ and I’ll return to some of these in amoment.

But the most straightforward means of inferentially extracting numericalinformation from a stretch of differential equation syntax is to apply a standardnumerical method like the Euler’s rule routine outlined in Essay 2.18 On this basiswe can grind out a sequence of rounded off numbers linked to a characteristic stepsize Δt. Pictorically, these computational steps correspond to successively filling insquares on a piece of graph paper whose dimensions have been set at the mesh sizeΔt. Plainly, if we pick squares that are too large, our graph will stop suppling anaccurate portrait of avian flight after a small number of Δt computational steps. Butwe can normally presume--I’ll come back to this “normally” qualifier later on--thatif we choose the Δt mesh size small enough, the computational path inscribed on ourgraph paper will resemble the bird’s actual trajectory closely for a reasonable spanof time. We also recognize that minute accumulated errors will eventually spoil ourresults, depending upon the size of Δt . On this same basis, we normally presumethat, if we could only reach the pointwise limit19 of all of these ever-refining graphs,we would asymptotically reach a continuous curve that exactly copies our goose’sflight path (presuming that we had, indeed, correctly registered all of the pertinentphysical factors within our modeling equations).

But we shouldn’t pretend that we can actually reach it. In terms of brutecomputational accomplishment, we can rarely calculate the exact features of theselimiting curves through our numerical reasoning methods. We merely obtain a set ofimproving approximations, depending upon the computational effort we apply to the

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vector field laid out by differential equations

problem. Through such methods, the exact details of the twists and turns within ourdifferential equation’s “true solution” may remain as far from our ken as the goose’sflight itself. But it is, nonetheless, comforting to know that the limit curve is there(if, indeed, it is–it may not be, as we’ll also see) because that assurance provides uswith an informative portrait of how ably the computational patterns we canconcretely lay down on a piece of graph paper correspond to the target behaviorsthey hope to capture. Yes, we can’t always calculate exactly what we want, but wecan perform helpful computational activities whose relationships to our unattainablegoals can be clearly understood. If we’re lucky, we may be able to establish ε/Δtestimates that supply us with the margin of error ε in which our graph paperdisparities fall.20 In this manner, we can frame an informative portrait of how ablyour computational skills relate to a naturally unfolding physical process (a goose’strajectory), rather as we can amicably estimate our pinball tracking skills. And sothe limitations revealed don’t seem particularly troubling, for our “informativeportrait” assures us that, although the desired flight curve usually lies at a“transcendental remove” (= would require infinitely many computationalrefinements to reach21) from our actual computational capacities, that removeremains mild in the sense that (a) we know how to reduce the degree of “remove”by increasing the Δt refinement of our computations and (b) we possess a clearunderstanding of how the target curve relates to our actual computational capacities,where it is likely to remain close and where it is likely to diverge.

Here’s an alternative manner of picturing this form of mildly transcendentalrelationship. Differential equations in two variables directly lay out clouds ofinfinitesimal tangent vectors in their own right (i.e, they establish a vector field). From these little arrows, novel curves can be assembled by connecting the arrowsend to end in the manner of a child’s fill-in-the-dots puzzle. FIG: VECTOR FIELDLAID OUT BY DIFFERENTIAL EQUATIONS But we recognize that, as long asthe arrows we draw remain of finite length, their endpoints can be connected in awide variety of ways. So it is a matter ofconsiderable transcendental assurance to learnthat, barring the intervention of singularities, sucharrows will connect together uniquely at theinfinitesimal level, even if we can never carry outthe required connect-the-infinitesimal-arrowsoperations ourselves.

Considerations of this nature supply us withwhat I earlier called a portrait of our computational position within nature. This

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ascertaining our computational place in nature

assessment warns us that it is unlikely that we ever be able to track our goose withprecise exactitude (we can’t always calculate what we want), but is nonethelessreassuring in that these incapacities remain mildly transcendental. We can still feelthat we grasp the processes involved within avian flight capably enough. Why? Due to our trust in the intervening differential equation model. In presentcircumstances we may feel that these modeling equations have correctly captured atan infinitesimal level the physical processes that generate our goose’s various flightpatterns, even if practicable algorithmics will never supply us with perfectrealizations of such curves on a finite scale.

If this new portrait of mathematical endeavor maintains itself (I’ll call it “anaive faith in differential equation picture”), we will have greatly improved ourestimation of our computational place in nature in comparison to Descartes’ gloomyuse-it-when-you-can opportunism. For the first time in mathematical history,22 ournew limit-tolerant attitudes render the prospects of a genuine mathematical optimismtruly viable, viz., the thesis that every process encountered in nature can be fit to anappropriate set of differential equations, with suitable side conditions adjoined. Indeed, Euler’s infinitesimal equations for an incompressible fluid appear as if theycan resolve Descartes’ pipe problem without much fuss.23 To be sure, thesecomputations are often so daunting that they can never be carried out to anacceptable degree of accuracy within a reasonable span of time. Nonetheless,the“mathematical optimist who plants her trust firmly in the soil of differentialequations can reasonably claim that nature remains “fully understood” at theinfinitesimal level within all of itsworkings, even if we lack a robust abilityto extract reliable numbers from suchequations through reasoning tools availableto us within mathematics proper.24 FIG:ASCERTAINING OURCOMPUTATIONAL PLACE INNATURE

Insofar as I can see, attitudes veryclose to this naive trust-in-differential-equations picture continue to animatepopular conceptions of how mathematicalphysics operates within contemporaryphilosophical circles, including the off-handed manners in which Theory Tenthusiasts blithely speak of the unblemished “models” attaching to their “laws.”25

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But we’ll later find that serious descriptive blemishes tarnish this simple faith (whichI’ll often label Eulerian26 in the sequel) and its attendant descriptive optimism mustbe replaced by some more sophisticated assessment, lest we become forced back tosome opportunist assessment of a liberalized Cartesian stripe. In the final analysis, Idon’t believe that any of us can competently resolve these issues at the currentmoment and in the sequel I will merely outline the future prospects for improved“optimisms” and “opportunisms” as they presently appear at our current stage ofscientific understanding.

However these issues sort out, we should recognize that a truly significantenlargement within mathematics’ descriptive obligations has entered our conceptuallandscape. Descartes and his contemporaries had presumed that mathematicsmerely serves as a fount of trustworthy reasoning tools that we should try to fit tonature’s own behaviors whenever we can. But in summonsing up mildlytranscendental realms in which differential equations can draw unfamiliar curveswithout direct human assistance, mathematical thought now offers landscapes inwhich the viability of applying a reasoning procedure of a computational stripe canbe assessed. Consider the three central ingredients in the picture of goose flightcalculation presented above: (1) the goose’s actual flight curve, (2) our differentialequation’s true solution modeling of that flight and (3) the numerical estimates wecan actually obtain from concrete numerical reasoning rules, based upon a choice ofstep size and roundoff procedure. To Descartes, mathematics only suppliesreasoning rules of type 3; it supplies no assessment of how short of their type (2)targets these computational rules might fall. Nor does Descartes anticipate thequestion (which often emerges within applied mathematics’ subsequent history)whether the mathematically designated targets (2) of our (3) reasonings actuallymatch the physical circumstances (1) in which our interests originally lie. Interminology I will frequently employ later, assessments of type (2) intrinsically relyupon generic “pictures” of how the formulas in our calculations relate to theirintended targets, e.g. that our goose’s flight path can be represented by a smoothmathematical curve. And sometimes these “pictures” themselves fall into error. For it often happens that nature presents us with explanatory mimics in which ourconcrete calculations (3) appear to be transcendentally supported by “true solution”constructions of type (2), when, in fact, some more sophisticated and considerablydifferent construction (2*) captures the physical circumstances at hand (1) moreaccurately. Thus we may initially fancy that the type (2) targets of our calculationsshould be smooth curves, but a closer examination of nature shows that such apicture is erroneous and that fractal trajectories (which are nowhere smooth) serve

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as better (2*) representations of our targeted circumstances than the smooth curves(I’ll supply an apposite example involving a stream of beer later in the essay).

The task I have described as assessing our computational place in natureattempts to develop a story of how ably we can keep these sundry ingredients inalignment or not. As such, the descriptive chore doesn’t strike me as markedlydifferent in conception from the ecological assessments that biologists offer withrespect to frogs and mayflies, beyond the fact that we possess swifter resources foradjusting to real world adversity than they do. With our frog, for example, the flightpaths of its intended prey (1) offer the frog a strategic opportunity (2, outlinedbelow) which it can exploit, to varying degrees of capture success, in its muscularreactions (3) to the appearance of the fly. Because of the centrality of strategywithin these alignment concerns, mathematical thinking of some sophistication onthe part of biologists is sometimes required to carry out these ecologicalassessments properly, even with respect to lowly frogs and insects. The centralthesis of the present essay is simply that any conception of “naturalism” that doesn’tinclude these ecological concerns as part of its descriptive brief shouldn’t beregarded as a proper naturalism at all. When we turn to the natural histories of human beings, issues of strategicadjustment become especially prominent, because as a species we are particularlyadept at plastically refashioning old reasoning routines for more refined purposes,by cannily borrowing effective reasoning policies from unexpected sources. So weneed to ask ourselves: in what ways does thinking of a recognizably “mathematical”character assist us in these strategic adjustments and when does it not? Any answerwe provide must be provisional, I think, for we do not yet know the extent to whichnature will ultimately cooperate with our descriptive wiles.

(v)

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two strategies for potentially improving a computation

Accordingly, our computational limitations frequently force us to cobbletogether reasoning gambits that operate according to subtle forms of alternativestrategization. The Euler’s rule reasoning we applied to our goose’s trajectory isconceptually straightforward but tedious (an extremely small step size Δt may beneeded) and highly subject to unanticipated errors (I’ll supply an example below). The infusion of a bit of extra information and a mapping to some alternative settingcan help immensely. Simple illustration: suppose we know that our goose usuallyflies in approximate circles around its nest, keeping a watchful eye on the young‘uns there. If we reexpress our governing equation in elliptic coordinates centeredon the nest, a modification of Euler’s method to suit these altered coordinates maysupply reliable results at a much larger step size. But if our fowl instead displays atendency to wobble back and forth periodically, we may want to decompose itsflight into a “slow time” sinusoidal trend complicated by various “fast time”perturbations. And so on. FIG: TWO STRATEGIES FOR POTENTIALLYIMPROVING A COMPUTATION

In such circumstances, we look around for a good strategy for addressing ourcomputational task in a tractable way and sometimes these alternative modes of“physics avoidance” can prove unexpected and subtle (we sees our strategicopportunities and takes ‘em). But allied considerations apply to mayflies, frogs andevery other adaptive creature. The basic subject of biomechanics assesses thebehavioral opportunities latent within an ecological niche and considers how an

organism might avail itself of these capacities (often in imperfect ways, as Stephen

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Hamilton-Jacobi factoring

J. Gould’s many writings on the subject have stressed27). Articulating the availablestrategies is a formal matter, sometimes requiring fairly sophisticated mathematics,as we’ll observe with respect to our frog below.

One of the most promising techniques for improved computation lieswith the general ploy of factoring a complex behavior into simple components,which the mathematician R.J. Walker explicates as follows:

An aspect of the interplay between analysis and synthesis in themathematical investigation of possible kinds of object is the attempt todecompose the objects under investigation into simpler ones which do notdecompose further, investigating the indecomposable objects first and thenbuilding up all the others. This going back to simplest objects is a reductionprocess, and hence one calls the indecomposable objects "irreducible"28

The basic prototype for this behavior is the manner in which the integers decomposeuniquely into primes, and the roads to many deeptheorems travel through that representation. Sosuccessful was this policy that Gauss, Kummerand their successors artificially introduced missingprime factors into other parts of number theory sothat imitative decompositions could be pursuedthat likewise led to splendid results (we’ll returnto those “extra ingredients” later). Withinmechanics, this book’s favorite example–theFourier decomposition of a vibrating string intostanding wave eigenfunctions–represents afactorization policy par excellence (I’m sure that my readers are thoroughly sick ofstrings by now, but they illustrate so many methodological morals). Generalizingupon these advantages in a more abstract manner, Hamilton-Jacobi technique withinmechanics asks whether the complex dynamical movements on a base manifold canbe factored into a set of simpler behaviors in which various descriptive variablessteadily wind their individual courses around their own tori (= doughnuts, if youprefer). FIG: HAMILTON-JACOBI FACTORING In most cases, the originaltrajectories of a problem are too entangled to permit such an unraveling, but when aHamilton-Jacobi factoring proves feasible (or even close to feasible), we ipso factolearn a great deal about the target system’s qualitative behavior (e.g., that it is notchaotic).

Mathematicians have found that the best way to understand a strategy such asfactoring is to locate a nearby domain in which the technique supplies perfect

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Fourier factoring of stringbehavior

a rattleback and its attractive basins

answers and to then relate present circumstances to this better setting. Thus thebehaviors of the algebraic ring based upon 15 are easier to comprehend if wecompare them to the unique factorization “ideal” in whichthe fused factors “inside” 15 get pulled apart. In the“reasoning pathways outside of the home country” analogy,I have often employ, situating our original 15 ring withinthis wider landscape makes it easier to comprehend theadvantages and obstacles encountered within topography ofthe home country.

The imperfect volcanos” analogy of Essay 6 providesanother geographical illustration of this point. Even if a reallife landscape fails to contain perfectly conical mountains,we can still employ their hypothetical locations as firmcoordinate landmarks from which we can usefully “factor” the surroundinglandscape. And we will want to do this even in very abstract settings: the weird

reversals of the rattleback toy can beunderstood as behaving like a mountaingoat that erratically jumps from oneattractive “valley basin” to another in itsphase space as friction gradually slows thecontraption down.29 FIG: ARATTLEBACK AND ITS ATTRACTIVEBASIN

Indeed, our “perfect volcanos”appeal might conceivably possess biologicalsalience as well. Maintaining fixed radial distances from a landscape’s impliedvolcanic cones might provide a grazing animal with a better foraging strategy thansimply gobbling up desirable foliage wherever it seems abundant. Why? Becausethe latter policy might ultimately strand the creature within a region bereft of foodstuffs, whereas adhering to a fixed elevation may more reliably lead to continuedresources, given that vegetation around a volcano cone is typically correlated withelevation and soil chemistry. It is fairly easy to concoct visual systems--such as ourown!--that immediately “see” the perfect cones within a landscape, filtering awaythe topographical irregularities as unwanted noise.30

Our fly-catching frog example is contrived to illustrate how another form ofembedding within a richer structural landscape helps us identify the availability ofparticular strategic opportunity. FIG: THE STRATEGY BEHIND THE FROG’SOPPORTUNITY If the insect victim’s reversing curves are very tight and their

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the strategy behind our frog’s opportunity

trajectories remain approximately coplanar atthese reversing points, the escape paths can benicely projected onto two-dimensional curveswith cusp singularities as shown. Mathematicians advise that such curves can beapproximated by so-called “Puisseaux series”whose beginning terms take the form of fractionalpowers--e.g., terms that look like ax1/3. Incomparison, regular portions of the flight (that is,away from the turning point singularity) need tobe approximated by the power series expressions familiar from elementary calculuscourses, of the non-fractional form a + bx + cx2 + ... So it frequently happens thatwe can approximate the escape path with a smaller number of terms near asingularity than within smoother locales where regular power series expansions areneeded. This representational simplicity offers our frog an exploitable opportunityfor estimating where its victim is likely to travel: wait for its tight turning points andestimate, as quickly as possible, the strength of the undetermined coefficient withinthe leading term ax1/3. In contrast, our amphibian friend would need to study theflight path far more intently before it could supply the larger number of estimatesrequired for a power series-based tactic.

Of course, it is highly unlikely that our frog’s brain will actually calculatecoefficients and fractional powers directly, but it might easily mimic the requisitedeterminations through employment of a look up table stored as trained muscularroutine. Nonetheless, our mapping-to-a-cusp-singularity story admirably explainswhy evolution would have fitted out our frog with a fly catching routine of thischaracter (workers in biomechanics commonly isolate the range of optimalenvironmental strategies first and then explicate any departures therefrom byreference to the animal’s genetic and physiological limitations). In short, we can’tproperly understand adaptive behaviors unless we have diagnosed the strategicenvironment in which such behaviors emerge. But the relationship of profitablestrategy to opportunity is essentially a mathematical question: the outstandingbiological mysteries of why certain animals forage as they do are frequentlycombinatorial in character.

The fact that our frog can readily implement a somewhat sophisticatedcapture strategy while not understanding in the least why the technique works willprove important in section (viii).

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advantageous singularities

Note that the key to our frog’s strategy lies in the behavioral simplificationsthat arise in the vicinity of a singularity. In an allied vein, Felix Klein31 (illustratingsome central ideas of Riemann) pointed out that the best route to understandingwater flow within a bath tub is to likewise start with its singularities: that is, theinlets and drains where the water comes in and out. Even if there don’t appear to beany “natural places” where such singularities might be situated, we might add suchsupplements as useful extension elements. For example, the internal behavior of aloaded, linear drumhead can be computed by adding up a passel of influencefunctions that tell us how singular point disturbances around the interior affect it(that is, if we clamp only point A with a unit weight, the interior will distort inmanner Γ). However, we can often obtain better results involving fewer nodes ifwe imaginatively “clamp” our drumhead at singularities lying at a fictitious distanceaway from its actual edge, allowing us to match local boundary conditions at thetrue drumhead edge by arcs found within the influence functions belonging to ourimaginary extension elementclamping points (this ployrepresents the centralstratagem behind boundaryelement technique). FIG:ADVANTAGEOUSSINGULARITIES Out-of-country representational tricksof this ilk are rampantthroughout appliedmathematics, as the strangepolicies discussed in Essay 8 vividly illustrate. We shall soon see how theinvocation of other sorts of out of country singularity allow us to unravel some ofthe great historical mysteries of computational failure.

More generally, nineteenth century mathematical discovery has taught us that,from the point of view of pithy data registration, singularities are often ourinformational best friends (as are whatever secret factoring capacities may lie athand). Among many other reasons, through their compactness, we can easily movesingularities round easily and through such means easily recognize how a targetsystem might be reconfigured to achieve better controllability (the complexsingularities revealed through a Laplace transform serve as an electrical engineer’sbest friends in circuit design). Naive thinking about “meaning” within applied

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shock wave singularities

Riemann

mathematics, however, invariably regards singularities as bad things, because theyrepresent circumstances where one’s modeling equations break down, by blowingup to infinity or otherwise becoming “meaningless.” But that represents a

shortsighted view of how descriptive terminologycan supply an “informativeness.” Theemergence of a blowup singularity can helpfullywarn us that we needn’t register enough pertinentphysics within our original modeling proceduresand that we should look to further physicalconsideration for help. For example, the shockwaves that form around airplane wings emerge inthe standard modeling equations for a gas asoutward descriptive inconsistencies that credit

the same regions of gas with incompatible velocities. FIG: SHOCK WAVESINGULARITIES Rather than rashly concluding that our initial modeling hasutterly failed at this point, Riemann (that great champion of singularities in general)realized that their appearance supplies a helpful signal that a bit of supplementarythermodynamic thinking needs to be applied to the problem. With this unexpectedinfusion of additional data, we find that we can continue to plot how these newshock wave singularities will move through the gas, employing exactly the samemodeling equations as before (but a broader notion of “solution” in the manner ofEssay 8). Sometimes a simple warning is sufficient to inspire corrective reasoning(“Halt! There’s a bear in this cave,” for example). Riemann’s“let’s tolerate the singularities” recommendations provide thebest policies to this day for understanding what occurs whensupersonic air rushes over an airplane wing. FIG: RIEMANN

At this point, we confront some rigidified assumptionswith respect to word meaning that create great confusionwithin pilgrimages such as ours. As detailed in Essay 6, manyphilosophers improperly presume that terminologies firmlyattach to the external world in uniform ways within appliedscience, allowing canny methodologists to completely foresee what every “kindterm” employed within a discipline will be like from the subject’s hypotheticalmoment of Theory T positing. This static conception of a discipline’s range of vitalproperties fails to render justice to the complex manners in which the interiorequations of a system interact with their side condition environment to generate

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descriptive quantities of great strategic importance (again, the tonal characteristicsof a confined violin string supply a sterling illustration which will be furtherdeveloped in section viii under the heading of Sturm-Liouville problems). In suchcircumstances, the only preexistent “names” for these vital quantities may stem fromthe mathematical landscapes in which the relevant reasoning opportunities are mostclearly marked, rather than arising from simple grammatical recursion over thevocabularies we employ to formulate the differential equations themselves (mostTheory T enthusiasts make this mistaken presumption). We’ll later find that theselandscapes of first discovery may lie very far away within Greater Mathematicslandfrom the applicational circumstances we initially confront.

The chief semantic moral within these remarks lies with the followingpragmatic observation. Notable advances in science often occur when someonenotices a nifty reasoning technique employed in field A and decides to try out thesame moves within field B, despite the lack of any evident connection between Aand B. Sometimes, after a bit of corrective tinkering, these coarse inferentialborrowings open the doors to bountiful new results within B (Oliver Heaviside’swork in linear differential equations supplies a remarkable paradigm for thisphenomenon32). But any advance in reasoning capacity of this nature automaticallybrings the vocabulary employed into tighter alignment with real world data, albeitoften in very complex ways. The same remarks apply to the seemingly fictitiousextension elements that we may have employed as way stations along the pathwaysof our newly opened inferential policies; sometimes these improved practicalentanglements wind up parking unlikely bits of syntax over important physicalcharacteristics.33 In exactly this manner, the phrase “our modeling equations blowup at time tn” succinctly registers some very important events that occur within theair rushing over an airplane wing, despite the fact that original authors of thoseequations had no advance inkling of these referential attachments whatsoever. Riemann’s stout faith that singularities should be nurtured and cultivated wasborrowed from his earlier experiences within complex analysis, inspired by thedescriptive opportunities that Cauchy had uncovered within that distant region ofinquiry (which I will outline below).

Such advantageous pilferings explain why I earlier claimed that“mathematics’s work is never done” in ongoing science. We simply cannot posit inone fell swoop all of the mathematics we’ll ever need within a projected science, asQuine and his comrades presume, for, like Riemann, we should continually consultgeneral mathematical experience as a fount of useful strategic ploys, often of anunexpected nature, stemming from structural arenas34 far removed from the task at

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Putnam

hand (e.g., when we first learn the advantages of factoringby experience with prime numbers and then transfer thesecomputational advantages, after suitable tinkering, togarbage can lids). As these strategic borrowings becomegradually fine tuned to obtain practical goals with greaterefficiency, the descriptive hold of our words upon on theworld tighten, whether or not we recognize how theimproved descriptive encoding operates. After all, we cansometimes achieve nearly optimal linguistic improvement through sheer trial-and-error tinkering, without enjoying any better understanding of why our computationalploys succeed than our frog possesses with respect to his admirable insectprocurements. I shall return to this important observation in a moment.

Hilary Putnam and Richard Boyd35 once claimed that “the terms of a maturescience typically refer,” indicating a presumption that, over time, predicativeexpressions in science will evolve to registering information about the world insimple predicate/natural kinds pairings (e.g., that “is 12o C” will eventually linktightly to some property involving mean molecular kinetic energy). FIG: PUTNAM Insofar as I can see, this claim is empirically false (it is certainly not true for “is 12o

C”) and there is no persuasive reason why linguistic improvements should inevitablyfollow such a simplistic course. To the contrary, the studies of fruitful forms oflinguistic labor in our other essays indicate that assigning individual sentencesstandalone informational content (e.g., interior equations versus the boundaryconditions with which they must coordinate) can prove quite problematic, due to theformal necessities of harmonious cooperation between the modeling ingredients. What Putnam and Boyd should have instead said is that, as a science matures, itsreasoning patterns and measurement techniques increasingly register correctinformation about the physical world, without making further simplisticpresumptions about the exact manner in which such information becomes encodedwithin the linguistic practices. Why have these authors presumed otherwise? Because they share rigidified prejudices about the semantics of language akin tothose we shall discuss in connection with Paul Benacceraf’s writings below.

Many of the would-be “naturalists” mentioned in section (i) appear to havenot appreciated the ongoing necessities of semantic fine tuning, having fallen captiveto Quinean fantasies of all at once Theory T postulation. Many of the reformistprojects that Maddy and I criticized in that section begin with the presumption that aworthy Theory T will be able to articulate all of its “physical properties” cleanly,allowing these authors to maintain that mathematics’ own range of posited elements

-24-

serves as a kind of shadow realm of independent entities that map in partialisomorphism to physics’ parochial “entities and properties.” This simple visionconjures up gallant naturalistic quests to rid science of its unwanted realm ofabstract objects, leaving behind only their pristine physical counterparts.

This quixotic vision does not remotely comport with the real life activitiesthat allow scientists to develop improving estimates of how “useful properties” ofthe external world can be captured within an effective set of reasoning policies. One of the chief reasons that humans are more swiftly adaptive to varyingcircumstances than frogs is precisely because they can readily borrow and retunereasoning stratagems developed for task A to become novel routines for achievingtask B, despite the fact that the relevant subject matters scarcely resemble oneanother beyond the formal consideration that both submit to some kind of factoringor condensation into singularities policy. Through these swift conceptual adaptions,thinking mathematically (in the sense of “pondering the strategies of a task in formalterms”) emerges as an important and distinguishing aspect of the natural history ofthe human race. Something has gone amiss with our “naturalism” if one fancies thatour normal policies for improving our linguistic control over nature need to becurtailed or purged.

This is why I earlier stated that Quinean talk of all-at-once positing does notcomprise a harmless idealization, but ignores the never-ending semanticrealignments that mathematical thought helps us find as we gradually enhance ourreferential grip upon the world around us.

Before leaving the topic of factoring behind, let me add a few remarks thatconnect our discussion to the chief themes of Essays 1 and 5. It reveals a furthersubtlety in what the phrase “mathematical thinking” connotes. Often it is helpful tofactor a time series (earthquake rumblings, stock price fluctuations, etc.) in one ormore dominant (often deterministic) trends alongside a stochastic part commonlylabeled as “noise.” The trend isolates the relatively stable dominant behavioraspects of the time series that we usually bear foremost in mind in planning asuitable reactive policy, whereas the noise warns us of the probable fluctuationsaround this mean for which we should compensate. But what sort of concept does“dominant behavior” represent? Does it “belong to mathematics” as traditionallyconceived? In some sense, the answer is “no,” because, as Essay 5 stresses,“dominant behavior” isn’t a notion that immediately falls within

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factoring a beer jet through a homogenized limit

mathematics’ ken, any more that a computer can readily sort out features of alandscape that we find obvious (“Can’t you immediately recognize Alfred E.Newman’s goofy visage within this photo?” we demand of a computer recognitionsystem; “No,” it replies, “not without a lot of subtle programming”). The problemwith our time series, roughly characterized, is that it contains a lot of jaggedcontours and we must apply some form of noise filter to extract the trend from itssurroundings. In practice, we generally achieve this “filtering” mathematically bysubjecting our time series to a rather strange set of operations that trick theunderlying mathematics into divulging its trend-like secrets. In particular, wetypically embed our data in a long sequence of strange extensions that allow ourhidden trends to finally shine forth mathematically in a final limit. For example,suppose we consider a stream of beer gushing from a cask that, intuitively, trendsleftward, yet which is also subject to Brownian motion (= its component moleculesget randomly jiggled at random moments by their environment). FIG: FACTORINGA BEER JET THROUGH A HOMOGENIZED LIMIT Intuitively, we regard thatupward trend as an important property of the fluid stream and will wish to utilizethis information in sagely deciding where a bucket should be placed to catch ourbrew (whereas the breadth of its mouth is determined by the noise factor). How dowe do this? Let’s start by considering a short range of four molecules within ourstream, as indicated on the left of the diagram, modeling the random environmentalcontributions by little stochastic arrows as indicated. We may fancy that we detecta trend in this pattern but untutored mathematics cannot. Instead, mathematiciansprogressively extend our four molecule chain to a sequence of ever larger models

-26-

involving an increasing number of molecules subject to similar stochasticdisturbances. These enlargement ae continued until they approach an infinitepopulation limit called a Weiner process whose streams now travel in unpicturablefractal trajectories that generally fail to possess well-defined tangents.36 It is only atthis stage that the trends they follow can be mathematically recognized, in the samefashion that the gaming means and variances pertinent to casino activity cleanlyemerge only within infinitely large populations of gamblers. Extension techniquesof this type represent classic exemplars of the asymptotic homogenizationmethodologies that allow various submodels in science to communicate with oneanother, as explained in Essay 5. Only in these strange limits will our “dominantbehavior” trends emerge from the enveloping stochastic mists as clean factorizationparameters.

Although the applications of a specific homogenization procedure surelyrepresents a “mathematical operation” in any reasonable sense of the term, it is notfully evident that its interpretation as a filter that brings forward a vital hiddenproperty of fluid stream qualifies as equally mathematical–it strikes me as asomewhat unnatural trick that happens to represent the best construction that presentday mathematics can provide for capturing how important quantities in natureempirically relate to one another upon different scale levels. But I’ll return to thistheme after a few intervening sections, at the point when we start ponderingmathematics’ descriptive limitations in a more pessimistic vein.

(vi)

Accordingly, we often range far afield within Greater Mathematicslandsearching for strategies that we can usefully borrow for descriptive purposes. Infact, our inherited recognitional skills force us to scour even wider landscapes forhelpful clues due to psychological limitations that arise as significant aspects of ourbiological parentage. Echoing the themes of human limitation characteristic of manynineteenth century authors (in which I fully concur), Hermann Lotze writes:

Now a tool must fulfill two conditions, it must fit the thing and it must fit thehand....[T]he human mind ... does not stand at the center of things, but has amodest position somewhere in the extreme ramifications of reality. Compelled, as it is, to collect its knowledge piecemeal by experiences whichrelate immediately to only a small fragment of the whole..., it has probablyto make a number of circuits, which are immaterial to the truth it is seeking,but to itself in the search are indispensable.37

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our avatar

“proof” through areal rearrangement

We do not “stand at the center of things” in the sense that we possess neither theperfect observational capacities nor the perfect inferential tools of a well-equippeddeity but must cobble our way forward by elaborating upon aroughly hew, though effective, set of initial computationalcapacities. So let us begin where all of our human strategicdeliberations begin, within the complex and hard-to-chart reasoningroutines that our distant forebears developed for the sake ofefficient hunting and foraging. FIG: OUR AVATAR In so doing,we, like many of our animal cousins, have developed admirablecapacities in, e.g., geometric anticipation (when objects A and Bcollide, where will their contact points lie?) whose achievements challengecomputational algebraic geometers to this day. Framing and passing alongfoundational capacities of this power requires long stretches of developmental timeand prolonged and sheltered childhoods that allow for finely tuned learning. Wherehumans differ from most of their cousin animals lies in their remarkable capacities totransfer these reasoning patterns to applications that have not been contemplatedbefore. Indeed, the swift expansion within evolutionary time of human reasoningcapacity seems explicable only through this plastic reallocation of fixed algorithmicresources, for everything we know about the evolution of human intelligenceindicates that our brains’ remarkable abilities to cobble together ancestral reasoningschemes for novel purposes lies at the center of our abilities to adapt to newenvironments quickly without needing to rely upon the slower processes of geneticadaptation.

But why do we recognize decompositional patterns more easily in somecircumstances rather than others? Answer: our aboriginal reasoning systems arrive

better acclimated to certain reasoning recognitions ratherthan others. Consider, in this light, the celebrated imaginedmotion verification of the Pythagorean theorem. If wedirectly inspect the three squares in the top of our diagramdirectly, we fail to detect any compelling reason thesummed areas of the little squares should coincide with thearea of the large square on the hypothenuse. FIG:“PROOF” THROUGH AREAL REARRANGEMENT Butif we study how these areas will remain confined undermotion when embedded inside the large square in thebottom figure, we immediately see that an additive identitymust arise (as the repentant slave trader in the old hymn

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psychological advantages of a transferred representation

a cannon ball to clothesline reasoning transfer

articulates, “I once was blind but now I see”). Why are we able to immediatelyreach a conclusion that eluded us above? Plainly, situating the imagined motionwithin a constant reference square allows our ancestral theorem anticipators to keepbetter track of volumetric relationships.38

In a similar vein, the curve on the left possesses some unusual geometricalproperties related to the manner in which its tangent lines wiggle about as wetransverse the figure, but these traits scarcely seem salient upon normal inspection. However, if we transfer the same curve into a representation within so-called linecoordinates, whose tangent lines have mapped over to points, we can immediatelysee exactly where the curve’s oddities lie–in the sharp singularities that have madethemselves manifest. FIG:PSYCHOLOGICAL ADVANTAGEOF A TRANSFERREDREPRESENTATION Why? Presumably because our inheritedmotion analyzers keep better track ofposition change than turning anglewhen we circumnavigate a figurevisually. Such examples explain whymathematicians often declare that a well-chosen transfer of a problem into a novelallows them to “look at the problem with a fresh set of eyes.”

Lastly, the caveman example of Essay 6 comprises a sterling illustration of afruitful transfer between a computational routine well adapted to modeling thetemporal development of a process over to circumstances where an equilibriumconfiguration is wanted (no registration of time appears at all). FIG: A CANNONBALL TO CLOTHESLINE REASONING TRANSFER hese formal similaritiesallow computer scientists to take an old marching method package off the shelf andreassign its steps to shooting method purposes (although we must tinker with the old

routines by adding on a further stretch ofmonitoring for self-consistency). Despite thesyntactic surface resemblance of the inferentialsteps followed within the two routines,semantically one deals with information in quitedifferent ways within the two circumstances. Historically, it took a fair amount of time beforemathematicians fully appreciated how different thetwo computational situations were, given the fact

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that the shooting methods begin life as adaptive offshoots of simple marchingmethod routines.

The long and short of this discussion is that even our most successfulstretches of reasoning will forever carry traces of their aboriginal origins and thisimpressive but compromised heritage remains the bedrock of wherever we wanderin constructing strategies for dealing with nature effectively. In an allied vein, ErnstMach very much hoped that physics might wean itself from doctrinal dependenceupon the mechanical ideas of substance that we have inherited from our ancestors,but simultaneously conceded that our improving descriptive capacities must alwayspiggyback upon our hunter-gatherer skill set:

But the natural philosopher is not only a theorist, but also a practician. Inthe latter capacity, he has operations to perform which must proceedinstinctively, readily, almost unconsciously, without intellectual effort. Inorder to grasp a body, to lay it upon the scales, in short, for hand-use, thenatural philosopher cannot dispense with the crudest substance-conceptions,such as are familiar to the naive man and even to the animal. For thehigher biological step, which represents the scientific intellect, rests uponthe lower, which ought not to give way under the former.39

Such considerations force us to ask several questions. Within what areas of farawayendeavor are we apt to first recognize the prospects of a useful strategy, forpotential employment elsewhere? And, once we have hoisted ourselves up by ourcomputational bootstraps through such borrowing, how might we ratify, with anyassurance, that we have made wise choices in adopting these altered and transferredprocedures? Finally, through what means can we reform our descriptivevocabularies so that we feel that we have adequately captured nature’s ownactivities in terms that are inflected by our biological heritage? In short, how do wemanage to spin scientific gold from hunter/gatherer straw?

There are two major aspects to these questions, which we shall addressserially.

(vii)

The mere fact that we have located an effective strategy for exploiting thedescriptive opportunities latent in nature does not entail that we have properlyrecognized why such an inferential gambit should be trusted, anymore that beingable to ride an bicycle well insures that one appreciates the supportive physics

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Dr. Mike’s card guessing trick

behind the activity.40 These cognitive blind spots are also largely attributable tointellectual limitations inherited from ourhunter-gatherer ancestors. Consider thefact that we often find parlor tricksinvolving card guessing puzzling. Of acertain manipulation of this type, PersiDiaconis and Ron Graham write:

It’s a charming trick and reallyseems to surprise people. Okay. How does it work? Let’s start bymaking that your problem: Howdoes it work? You’ll find it curiouslydifficult to give a clear explanation. In twenty years of teaching, we haveyet to have anyone give a truly clearstory.41

How does one supply such a “clear story”? Let’s investigate a considerably simplersituation: a trick that I obtained from Dr.Mike’s Math Games for Kids website.25 FIG: DR. MIKE’S CARD GUESSING TRICK The subject is asked to select anumber from 1 to 100 and then asked whether her choice lies on seven profferedcards like the three shown on the left of the diagram. From these responses themagician “magically” extracts the correct original choice. On the right wetransparently witness the underlying story of the information processing that makesthe trick work: the subject has unwittingly supplied direct specifications of thesuccessive digits in her number as registered within decimal notation.

One figures out the murderer in a board game like Clue through an allied formof Twenty Questions enclosure. FIG: A CONTRACTIVE STRATEGY A wisestrategy for playing this game should squeeze in on the required answer via arefining set of questions as efficiently as possible, which usually requires that theinitial questions partition the available search space into blocks of roughly the samesize. Mathematicians say that such strategies operate through contractive (orcoersive) entrapment of the desired answer--we progressively pose inquiries that

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a contractive strategy

ultimately squeeze in upon a final answer as a “fixed point”(“Col. Mustard in the dining room with the lead pipe”). Sometimes we must balance our desire to reach a correctanswer as quickly as possible (i.e., employing the fewestnumber of questions) against the assurance that we alwaysreach a right answer even in unlikely situations andprobabilistic strategies for obtaining answers can operatemore swiftly if we either accept a few wrong answers ortolerate the risk of inconclusive cycling.26

If a well designed scheme of this nature is laid outclearly, in the manner of our explanation of Dr. Mike’scard trick, it becomes obvious that the steps in the process

are contractive: we can readily see why the progressive questions asked graduallysqueeze an already narrowed search space into yet smaller components. But onecan also play Twenty Questions or Clue utilizing a dumb methodology that provesredundant rather than coercive–children do this all of the time.

In this fashion, successful magic tricks call upon our intellectual heritage asself-improving aboriginal thinkers in two fashions: (1) the fact that, presented in asuitable disguise, the contractive character of a set of questions may allude us yet;(2) mapped into an alternative representational framework in the manner we appliedto Dr. Mike’s routine, the clear story behind the trick can become completelyevident. With respect to (1), the Great Tomsoni explains:

When people see a wonderful piece of magic, they try to figure out how it’sdone. They have avenues of thought and logic. The magician, just beforethe denouement or finish, must close all those doors. The only solution ismagic.27

By the “avenues of thought and logic,” he intends the collection of possible eventsin which the audience attempts to locate the magician’s manipulations, when, infact, they actually fall within some collection that they have not considered. Withrespect to the trick that Diaconis and Graham discuss, the “magical” concealmentinvolves disguising the continued preservation of information throughout a range ofmanipulations that the audience expects will destroy all data of the sort in question. Indeed, the “clear explanation” they supply for their trick relies upon mathematicalinduction: the fact that if a starting trait proves hereditary over a chain, then everydescendent within that chain will possess that trait as well. As such, the reasoningprinciple itself appears intuitive enough, but the unexpected ordering in the cardsthat they show must persist through all of the shuffles is not (it strikes us as very

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blowing up a curve to understand itssingularities

abstract). Did the proof we just gave ruin the trick? For us, it is a beam of lightilluminating a fuzzy mystery. It makes us just as happy to see clearly as tobe fooled. 28

To obtain this “beam of light,” the authors must map the concrete cardmanipulations within their trick into the purified settings of graph theory before theappropriate “Ah Ha! Now I see it” recognition gets prompted. So we confront atwo-way street here: we can often construct mystifying magic routines from easy-to-follow inheritance chains through purposefullyhiding their contours under all sorts ofpsychologically distracting clutter.

Here’s another example of the increasedunderstanding that comes with an artfully selectedtransferred mapping. Considered in its own light,the singular point at the center of the 2D bow tiecurve on the bottom plane of the illustrationexhibits puzzling behaviors with respect to itspoints of intersection with other geometricalfigures, but the proper rules become immediatelyclear if we blow up the point into the setting of a3D curve as illustrated.29 FIG: BLOWING UP ACURVE TO UNDERSTAND ITSSINGULARITIES Proceeding in this spirit, the great Italian geometers of the latenineteenth century untangled more complicated singularities in a very useful fashion,but doing so required shifting into spaces of a fairly high dimension. We’ll return tothe trustworthiness of these “transcendental” unfoldings soon.

Such ruminations uncover a second dimension to Lotze’s observation that ourinferential capacities are not located “in the center of things”: human understandingreflects ingrained limitations that we often surmount by mapping strategicallyopaque circumstances into settings for which our hunter-gatherer inheritance has leftus better prepared in terms of strategic appreciation (in the same manner as mappingthe tangent singularities of an algebraic curve into a point singularity setting provesintellectually revealing, despite the fact that, mathematically, both phenomenacoexist on a symmetrical par). In the history of science, practitioners havefrequently learned to execute an inferential strategy quite capably, long before atransferred mapping to another setting is located that renders the rationale behind itsunfolding strategically transparent.

-33-

But this situation is hardly satisfactory and intellectual progress will be betterserved if we can develop a better portrait of how our inferential strategies actuallyoperate, so that we are better prepared to understand or anticipate reasoning failureswhen they threaten. But these quests for greater strategic enlightenment (of thesame sort as we asked of our magic tricks) often carry us even further intounexpected parts of Greater Mathematicsland, for only there do we find the “Ah ha!Now I see” recognitions we seek. So let us briefly pursue mathematics’ grandestparadigm for “understanding inferential strategy” through mapping an originalproblem into a richer setting: the astonishing unraveling of series behavior thatCauchy and his followers achieved through imbedding the real line upon the planeof the complex numbers (or, even better, upon an appropriate Riemann surface).

Many of the greatest early discoveries within number theory and differentialequations were obtained by extracting infinitely long series expansions throughvarious techniques of formal manipulation (here “formal” means “obtained throughrearranging the formulas through ‘subtractions’ and ‘divisions’ in a naive manner”). Unfortunately, these otherwise stretches of fresh syntax display a disagreeablepropensity for supplying really rotten answers at unpredictable moments (e.g., theyblow up to infinity or approach correct values very, very slowly). Such bad spotsmake it very hard, in dealing with an unfamiliar equation, to know when onesconclusions have remained on track and when they have shunted into the realms offancy through plowing through a unforeseen bad spot unwisely.

Sometimes the reasons for these miserable behaviors are fairly evident, evento our limited hunter-gatherer understanding. If we extract a series expansion byformally dividing the formula 1/(1- x2), we can readily understand why the seriesconverges inside the (-1, +1) region (because we can easily find bounds on thepartial sums that gradually squeeze their results ever closer to a limiting value) butthese same considerations break down at the endpoints. However, we don’t findthis failure at -1 and +1 particularly surprising because the original formula becomesinfinite there (and hence shouldn’t possess a proper summation in any case). Butwhy does the analogous series derived in the same formal manner from the formula1/(1+ x2) break down at these same points, despite the fact that its parent formula iswell-behaved everywhere along the real line? To understand this strangehappenstance, Cauchy recommends that we look further out in Mathematicsland. In particular, we should ponder what transpires upon the complex plane (a bit later

-34-

Cauchy’s explanation of series breakdown

Riemann shifted Cauchy’s loci of strategic clarification to sundry Riemannsurfaces). Cauchy argued as follows. Given that conventional power seriesexpansions consist in additions and multiplications, our series computations will stillmake sense over the complex numberrealm, even if though our originaldifferential equations no longer carryobvious physical significance there. Considered from this point of view, wenotice that the partial sums remainconstrained by clear limiting boundsinside the shaded circular region butthat two bad spots appear at i and -i, ina manner completely analogous to the -1 and +1 blow up points for 1/(1- x2). FIG:CAUCHY’S EXPLANATION OF SERIES BREAKDOWN Accordingly, weshould doubt whether our expansion for 1/(1+ x2) can be trusted anywhere along thecircle of convergence that runs through +i and -i. But the real-valued points +1 and-1 lie on this bounding circle, so we have located a useful early warning signal ofproblematic series behavior by noting the obvious bad points on the complex plane.

Tristan Needham summarizes these discoveries: But how is the radius of convergence of a [power series for f(x)] determinedby f(x)? It turns out that this question has a beautifully simple answer, butonly if we investigate it in the complex plane. If we instead restrict ourselvesto the real line--as mathematicians were forced to do in the era in whichsuch series were first employed--then the relationship between [the radiusand f(x)] is utterly mysterious. Historically it was precisely this mystery thatled Cauchy to several of his breakthroughs in complex analysis (he wasinvestigating the convergence of series solutions to Kepler’s equation, whichdescribe where a planet is in its orbit at any given time).30

In short, to understand the special functions that naturally arise from the equationsof mathematical physics, their behaviors should be examined over a wider territorythan the real line. Mapping a function’s singularities upon the complex planegreatly helps us understand the otherwise mysterious computational failures that wewill encounter in attempting to compute its values employing standard expansiontechniques. For example, the topography of a Bessel function’s singularitiesprovides the skeleton key we require to appreciate the patterns of computationaladjustment that should be followed in computing the shape of a drumhead correctly

-35-

Cauchy

a “mildly transcendental” true solution

via series expansion techniques. Such phenomena are deeply entangled with ourearlier observation that stock philosophical pronouncements tothe effect that the singular behavior of differential equationsupon the complex plane “carry no physical significance” arepoorly conceived. FIG: CAUCHY

Presumably, if we were more godlike in our inferentialcapacities, we wouldn’t rely so heavily upon such “transferredsetting” considerations to reveal the strategic soundness of agiven computational gambit. But we are not omnilogical andare easily fooled by magic tricks and series expansions alike. The only remedy is to transfer our reasoning policies into avariety of alternative settings until we find one where theroutine’s strategic virtues and vices become manifest to blinkered intelligences suchas ourselves. But the virtues and vices of inferential technique generally trace tostructural reasons that transcend any specific subject matter (the same sievestructure that effectively locates a target card for us in a magic trick can be used tosort out defective parts in an assembly line). So a transferred setting that assists usgreatly within physics needn’t seem particularly “physical” in its own right.

(viii)

Let us return to Euler’s original sanguinity with respect to the capacities ofdifferential equation specifications pitched at the infinitesimal level; they could betrusted as descriptively accurate despite the fact that poor mortals such we can onlycompute crude numerical approximations to the more finely wroughtcharacterizations of nature offered within theequations themselves. A MILDLYTRANSCENDENTAL TRUE SOLUTION As we observed, it is only by trusting “mildlytranscendental” vehicles of this new ilk thatthe stirring thesis of mathematical optimismcan be plausibly defended: viz, that everynaturally occurring process can be accuratelycaptured within the webbing of mathematicalthinking (lacking such resources, poorDescartes could only subscribe to a more limited thesis of descriptive opportunism). However, differential equations prove every bit as prone to puzzling reasoning

-36-

contracting to a bogus limit

escape horns attaching to Euler’s method

failures as the series expansions just surveyed and demand some Cauchy-likeexplanatory portrait of underlying territory that allows us to strategically justify orcritique our reasoning gambits within this particular stretch of applicational territory.

Indeed, standard approximation techniques for differential equationssometimes misfire, often for initially mysterious reasons that can lead to engineeringdisaster. Let rejigger the differential equation that I (tacitly) employed in drawingour cannonball plots in an innocuous-seeming fashion. Based upon the ball’s initialvelocity when it leaves the cannon’s mouth, we immediately know its current fundof energy E0, as it is presently expressed in an entirely kinetic format in origin (i.e.,E0 = ½ mdy/dt|t0).31 Next, consider the firstorder equation that articulates the conservationof energy for our system that correctly claimsthat, at every moment, the sum of our ball’snewly acquired potential energy (measured bygmy) and its current kinetic energy will remainequal to E0: ½ mdy/dt + gmy = E0. But if weapply a standard numerical procedure to thisreplacement formula, we often obtain plots assketched in heavy black: cannonballs thatmysteriously levitate forever once they have reached their highest crest. FIG:CONTRACTING TO A BOGUS LIMIT No matter how finely we refine our gridlength Δt, we continue to get these idiotic results. What has gone wrong?

Well, we usually can’t know what a full solution to a differential equation willlook like before we begin to plot it, but by inspecting the coefficients of itsgenerating equation closely, we can often set bounds on what a worst case outcomemight look like in the short term. Suppose we select an initial instant and step sizeΔt and look carefully at the first modeling that gave us nice plots using Euler’s rule. We find that its leading coefficient places significant limitations on how far away

from the Euler’s method straight theequation’s real solution can wander withinthe span of a Δt interval. This datum(which is called an “apriori estimate”because it depends entirely on the form ofthe differential equation) allows us to drawin little horn-like regions in which theequation’s true curve and its Eulerianapproximation remain must trapped over

-37-

an imposter sneaks in!

the time span Δt. FIG: ESCAPE HORNS ATTACHING TO EULER’S METHOD Plainly, if we utilize a coarse time step Δt, it won’t require many iterations of ourEuler’s rule technique before the plots it draws run a significant risk of divergingwidely from their target curves. But we are still assured that if we instead pickshorter Δt intervals, the error within our approximating plots will stay withinmeasurable bounds. Indeed, as the grid length Δt is refined continuously to 0, theescape horns within our error estimates will enclose our equation’s true solutionwithin a coercive trap, forcing its limiting curve to coincide with a proper solutionto the differential equation as a mildly transcendental fixed point. In suchcircumstances, our Euler’s rule reasoning can be ratified as trustworthy as long asits step size Δt is chosen small enough.32 Such results are commonly calledcorrectness proofs and represent the appropriate analog to logic’s soundness proofswithin this setting.

But in the less nice situations of our revised cannonball equation, anunfortunate phenomenon overtakes our cannonball plotting at its turnaround point:the escape horn belonging to our modified energy equation opens up completely atthis juncture, no matter how finely Δt is chosen (nown in the jargon as a failure of aLipschitz condition). At this juncture our approximation method loses all of itscoercion and the resulting pathology allows imposter “solutions” to sneak into ourscenario and confuse our approximation method (which, after all, is too stupid todistinguish a “good solution” from a bad one). FIG: AN IMPOSTER SNEAKS IN! So this iswhere our second cannonball plotting went amiss:a ridiculous “solution” in which our projectilenever changes its elevation (which obeys ourrevised energy formula as ably as a normalprojectile) enters the premises and our numericalmethod loses its contractive handle on which ofthese “solutions” it should track.

In nice circumstances, it certainly looks as ifour contracting web of approximations is headedtowards a final curve as fixed point: indeed, thatappearance is precisely the phenomenon that inspired Euler’s original direct faith indifferential equations (and its attendant mathematical optimism). Yet we must becareful about trusting impressions such as this: Essay 8 reviews some celebratedcircumstances in the calculus of variation where the presumed fixed point simplyisn’t there, evaluated in conventional terms.33 And there are known examples of

-38-

Dedekind

differential equations that accept no true solutions whatsoever, despite appearancesotherwise. When we considered the coercive behaviors of the reasonings behind DrMike’s simple card trick or within a game of Clue, the contraction terminates after afinite number of steps that we can concretely verify. But an infinite number ofimprovements are required before our Euler’s method approximations fully squeezein on their fixed point targets and appearances are often deceiving in suchcircumstances. The paths of resolution that Richard Dedekind and others pioneeredrelied upon set theoretic reasoning and careful explications of “limit” to answerthese questions in a more trustworthy fashion, by replacing our intuitive pictures ofcurves and approximations by hard data on how coersive nets ofreal-valued n-tuples relate to one another through residual terms. FIG: DEDEKIND Allied set-theoretic thinking is needed tomonitor what happens when an algebraic geometer blows up apuzzling curve into vastly higher dimensions. At our presentmoment in developmental time, the infinitary landscapes ofmodern set theory serve as our most trustworthy courts of finalappeal in resolving basic existence questions of the sort sketchedhere. I get the uneasy impression that many philosophers do notappreciate set theory’s powers of strategic ratification as ably asthey should, for they commonly focus upon capacities (such as“providing ontological reductions”) that I regard as dubious or ephemeral. As aresult, set theory’s extended hierarchies appear suspicious to them, rather thanserving as the natural arenas in which elaborate contractive processes and alliedstrategies need to be examined. Thus arises the conviction that a well-intentionednaturalist should attempt to prune set theory’s excesses to less extravagantcontours.34 Empirically, however, this judgement seems wrong: our most refinedsuccesses in locating an intellectual landscape in which potential pitfalls of ourreasoning strategies can be cleanly recognized remains within the bailiwick of theiterative conception of sets. And the “extravagance” of its transcendental levels ofordinals should not be viewed as a naturalistic vice, because the very task ofassessing our computational place in nature requires a descriptive language in whichwe can adequately characterize the computational remove between nature’s ownprocesses and the humble calculations we can actually execute.

Ultimately, I think that this court-of-final-appeal privileging of set theoreticthinking rests largely upon its great successes in illuminating the otherwisemysterious puzzles of inferential technique. As we’ll soon see, plenty of othermysterious of reasoning remain within standard applied mathematics practice that

-39-

do these sand patterns reflect factoring modes?

have not yet succumbed to set theory’smollifying assurances. Perhaps these deepissues may need to be rethought someday butnot in the vein of the typical reformistprojects of would-be “naturalists.”

Unfortunately, set theory’s vitalcapacities for codifying mildly transcendentalrelationships between concrete calculationand their target objects have become obscured in philosophical tradition due toRussell’s and Quine’s dubious methodological fables of Ockham’s razor parsimonyand ontological reduction. But as Dedekind himself maintained, there is no evidentneed within mathematics for defining away interesting objects; in building up aunique factorization domain for the ring of 15 by collecting together equivalenceclass “ideals,” he believed that he was merely insuring that such a domain could beconsistently postulated (because of the concrete exemplar he constructs) and that itsinternal characteristics are clearly defined (because they can be induced from theproperties of the ring below). It is important for naturalists to recognize that, in thecore utilities canvassed here, set theory provides the natural vocabulary forarticulating the relationships that we lowly calculators bear to nature’s moreabundant collection of processes.

(viii)

Earlier we rejected the notion that the important traits attaching to a physicalsystem could be delineated by a simple recursion upon the “kind terms” employingits setting its basic modeling equations, observing instead that such quantities mustbe nurtured within the equation’s own mildly transcendental activities, as the interiordynamics of our target system interact with the environmental factors capturedwithin the model’s various side condition specifications. We especially consideredthe manner in which important factoring quantities such as the energeticallyindependent modes of a violin string. But this liberation from the straitened orbits of“kind term” definability raises an immediate concern, closely allied to thefundamental existence claims just canvassed: how can we reliably ascertain whenquantities seemingly spawned by a differential equation model possess genuineexemplars?

That this represents a non-trivial concern can be seen from the followingconsiderations. In 1787, the experimenter Ernst Chlandi had demonstrated that

-40-

Fourier, Sturm and Liouville

striking geometrical patterns could be produced by sprinkling sand on vibratingplates and touching them in one spot oranother.35 FIG: DO THESE SANDPATTERNS REFLECT FACTORINGMODES? Inspired by these discovries andFourier’s great work in the theory of heat,Charles Sturm and Joseph Liouville supplieda very clever general argument in the 1830'sthat a wide, but not universal, class of differential equations (conjoined to a suitableset of boundary conditions) always possess factoring modes of an energy conservingilk (technically called eigenfunctions), no how recondite from normal definabilitythese traits may prove to be (these discoveries are generally discussed under theheadings of Generalized Fourier or Sturm-Liouville analysis). Once these resultsbecame available, it became clear that, through touching his plates in the requisiteplaces, Chlandi had suppressed some of its secret Sturm-Liouville modes whileallowing the remainder to vibrate on. Guitarists are familiar with this methodology:if we lightly touch a string at its midway point, we drain energy from its fundamentaleigenfunction, allowing only the other overtones to continue ringing. The result is abell-like tone called a harmonic. Just so; Chlandi’s sand settled in the locales that hehad rendered stationary through suppressing some of the independent vibrationalmodes that usually participate in the plate’s behavior. Through such tests, laterphysicists experimentally verified the unsuspected presence of decompositionalmodes within a wide variety of target systems, exactly as Sturm and Liouville hadpredicted. Once uncovered, their great factoring capacities supply the vital skeletonkey that unlocks the seemingly unmanageable complexities of real life vibratorycircumstances. Many of the greatest successes within nineteenth century modelingendeavor followed this policy in one guise or other. Lord Kelvin commented:

Fourier's Theorem is not only one of the most beautiful results of modernanalysis, but it is said to furnish an indispensable instrument in thetreatment of nearly every recondite question in modern physics.36

As it happens, modern luthiers seek allied insight into the vibrationalbehaviors of irregular shapes such as the upper plates of guitars by sprinkling sandon them in hopes of obtaining cognate structural enlightenment. But in suchcircumstances, the search techniques central to Sturm and Liouville’s proof becomestymied due to a guitar’s asymmetric contours (we’ll learn why in a moment) andcannot verify any firm connection between sand patterns and secret repositories ofenergy. Indeed, to the best of my limited knowledge, it remains uncertain that the

-41-

“looks like a trap...”

stationary sand plots on a guitar face signify much of anything with respect toimportant internal characteristics. It remains entirely possible that factoringopportunities of the desired sort are simply not available within a guitar plate. Certainly, standard Sturm-Liouville arguments do not apply. FIG: FOURIER,STURM AND LIOUVILLE

As noted before, factoring qualities of this nature rarely remain the same asapplications alter, even when the quantities assuredly do exist (we’ll see why this isso from the proof sketch provided below). The factoring nodes available within aclamped homogeneous drum head (which obeys a two-dimensional version of thesame equation that applies to our string) are quite different from those found in aviolin string and represent the products of Bessel functions rather than sine waves. In most circumstances, Sturm and Liouville’s methods do not land on familiarfunctions at all, but squeeze out novel quantities such as the repositories ofunmusical energy that lurk within the clang of an unevenly manufactured (butsymmetrical) garbage can lid.37 To locate these recondite characteristics, Sturmand Liouville must search for them (like lions or unicorns).

It was within this setting that the central germs of what we now call settheoretic thinking entered into nineteenth century mathematics in a central way. Although the full details are too elaborate to recount here, the pathways of Sturmand Liouville’s reasoning hunt down their Fourier-like prey through severalnetworks of converging approximation of a markedly set theoretic character,prompting the author Hans Sagan to declare of the accompanying diagram (whichlocates the salient zeros of the successive eigenfunctions):“this looks like a trap andit is.” 38 FIG: “LOOKS LIKE A TRAP...” The abstract, formula-eschewingcharacter of their reasonings is now viewed as a bellwether of all modernapproaches:

The impact of these papers went well beyond theirsubject matter to general linear and nonlineardifferential equations and to analysis generally,including functional analysis. Prior to this time thestudy of differential equations was largely limitedto the search for solutions as analytic expressions. Sturm and Liouville were among the first to realizethe limitations of this approach and to see the needfor finding properties of solutions directly from theequation even when no analytic expressions forsolutions are available. 39

-42-

periodic motions of a string mode

Drumheads, strings and organ pipes obtain their respective modes through markedlydifferent processes than a violin string, so Sturm and Liouville could not directlyappeal to any evident quantities shared by the applicable systems. Instead, theyproceeded in the following general manner. Beginning solely with bare structuralfeatures of their target equations (how their coefficients are arranged) together withtheir exact manner of boundary condition confinement (in Sturm and Liouville’soriginal setting: fastened down everywhere and symmetrically arranged like arectangular plate or a circular drumhead), our authors must corral a layeredmenagerie of eigenfunction behaviors within a series of ever-tightening traps thatinsure that they will all conserve individual allotments of energy in the manner of aviolin string. To make progress on this front, Sturm and Liouville had to confinetheir attention to problems that naturally factor into one-dimensional circumstancessuch as a circular or rectangular plate (the symmetries of the boundaries areimportant for this purpose). Once this narrowing focus is made, they try to locatesome special initial conditions of their target systems in which the gizmo in questionwill move in a manner that periodically rescales its initial condition.40 In lessabstract terms, this is how a pure standing wave in a string behaves--we can initiallypull the string into any sine wave pattern we like and the string will maintain that

basic shape with smaller amplitude as thepatten wobbles across the spring axis,periodically shifting its energy betweenpotential and kinetic modes of storage (usuallywe picture this special configuration at itsturnaround point, when it is temporarilymotionless). FIG: PERIODIC MOTIONS OFA STRING MODE If Sturm and Liouvillecan locate a full set of privileged startingconfigurations of this character, they’ve ipsofacto located some important energypreserving modes, because in the linearcircumstances they consider, these modes lackany capacity to leak energy to one another. But how does Sturm (who is responsible forthis aspect of their proof) find these privilegedstates”? Essentially, by successivelysqueezing in on them (according to theirnumber of zero crossings) through a shooting

-43-

2D versus 3D problems

method variation on the cannon ball tracking scheme employed above. Havinglocated a system’s spectrum of eigenfunction behaviors in this manner, Liouvilleestablished the factorization part of the scheme: Sturm’s purist modes are completein the sense that any arbitrary solution to our target problem can be reconstructed asa potentially infinite sum of these special states.41

Again I stress that vital physical quantities such as these modes are ratified asnon-trivially applicable through existence proofs of a set theoretic character, ratherthan spun off as combinatorial operations upon a fundamantal vocabulary.42 In theabsence of such a proof, we lack comparable assurance that allied traits can belocated within the behaviors of asymmetrical objects such as guitar faces, evenwhen the “laws” (= differential equations) that govern the interiors of circularwooden plates and guitar faces are exactly identical. Why? We observed that Sturmand Liouville were forced to decompose their problems into one-dimensional piecesbefore they could reliably squeeze in on their eigenfunctions. Well, everyhomeowner plagued by animal invasions knows that it’s easier to capture a squirrelin the living room than a bat for the same topological reasons (I write from bitterexperience). FIG: 2D VERSUS 3D PROBLEMS And so it goes with trappingvibrational modes as well.

In this non-syntactic manner, many important physical quantities obtain theirnatural labelings, not as “kind terms” framed within the vocabulary in which theirsupporting posits are first articulated, but from the strategic technique that uncoversthem. As a result, we might nominate the quantities wehave just extracted as “the Sturm-Liouville qualitiesbelonging to this system” (although this description needn’tprove unique), but that designation reports upon the searchwe pursued throughout the structural hallways of greatermathematics (in this case, a bit of set theory), rather thanrepresenting a terminological fait accompli of the sort that“kind term” thinking anticipates. By approaching issues of definability in a carelessmanner, many contemporary philosophers have failed to acknowledge the crucialassistance that set theoretic construction plays in amplifying our notion of “physicalquantity” to workable proportions.

In presuming that they have captured genuine properties of their targetsystems within their ur-set theoretic nets, Sturm and Liouville (tacitly but rightly)abandon some cherished philosophical theses about our relationships to the world’sactual traits, following roughly pragmatic percepts. In particular, they rob worldlyproperties of many layers of “intensional” characteristics formerly attributed to

-44-

how Sturm-Liouville traits get their names

them. Even today within philosophical circles it is still commonly presumed that welearn about the world’s traits through the “meanings” of the descriptive vocabulariesare taught to employ, following some simple linkage between linguistic predicateand external property (such assumptions are latent in the “rigidified semanticthinking” we shall criticize in section (xi)). This makes us think of a property asenshrining some kind of “rule for classifying” that we can immediately follow, iflocated within favorable circumstances (it is hard to delineate the exact requirementsof such thinking precisely). But Sturm and Liouville find their special traits via asignificantly different route: they locate their modes in a mildly transcendentalmanner, as the system invariants that, allow us, inter alia, to carry out a long seriesof shooting method computations that provide an increasingly accurate fix on thetarget system’s true behaviors. Roughly speaking, Sturm and Liouville search fortheir target properties following the maxim “By such signs ye shall know them.” asthe fixed point terminus of practical calculations that indicate that they are closing inon something substantial.

With respect to a nomenclatural point made earlier, insofar as these newlydiscovered modes enjoy preexistent “names,” they derive from the region of settheoretic thinking in which they are found (labeled “Sturmland’ in the illustration),and are not directly generated by the terminology of the differential equation modelto which they apply. FIG: HOW STURM-LIOUVILLE TRAITS GET THEIRNAMES

And these remarks generalize significantly. A methodological theme of vitalconcern to the great philosopher/scientists of the late nineteenth century was thetask of providing underpinnings for what they often called “the free creativity of

science.” They (correctly) believed thattraditionalist expectations on what it is to“understand” a trait employed within scientificdiscourse were excessive and inimical to progress(for Mach and Duhem, these conceptualassumptions locked scientific thinking within acage of unproductive “mechanism”). To weakenthe hold of these barriers, they typically invokethe pragmatic value of science. For example, inHeinrich Hertz’ frequently quoted words, a

scientist merely needs to create “images” that can parallel natural behaviors in annumerically accurate fashion.43 Hertz, however, does not mention themethodological complications we have considered in this paper, but if he had, he

-45-

coordinate problem on a manifold

Uncle Jeff understood as a manifold

longitude coordinates, that ourresults eventually become

nonsensical. FIG: COORDINATE PROBLEM ON A MANIFOLD Suppose, forexample, that our animal flies in greatcircles embossed with a simple sinusoidalwobbling. Numerically, the magnitude ofthose side-to-side weavings apparentlybecomes infinite in the North Pole regionbecause of the manner in which latitudesand longitudes come together in the polarregions (in the jargon: coordinatesingularities appear). But no matter howwe alter the coordinate charts underlyingour compuations, we find that allied

difficulties will inevitably arise, due to the topological discrepancies between acurvaceous planet and our graph paper. The intuitive corrective, long familiar toancient mariners, is that we should periodically replace the charts in which wecalculate our bird’s flight with replacement maps centered upon new locales and tomodify apparent coordinate magnitudes by suitable adjustment factors (i.e., anappropriate metric).

To maintain the spirit of improved Eulerian optimism, we’d like GreaterMathematicsland to embrace gizmos that abstract away from our lowlycomputational woes and more ably parallel our goose’s actual circumstances. Indeed, the conceptual assembly we seek enjoys a fancy name: vector equation overa manifold. But how should such a notion become incorporated into mathematics’descriptive repertory while maintaining an acceptable level of precision? Theclassic remedy, pioneered by Dedekind, Weyl and others, employs equivalenceclass techniques and a general spirit of “by their signs ye shall know them”pragmatism to turn the trick.

Here’s the general idea. Suppose we want a friend to form a correctimpression of our absent Uncle Jeff. FIG: UNCLEJEFF UNDERSTOOD AS A MANIFOLD If wemerely show her an isolated photo or two, she willlikely form a somewhat erroneous picture due to theinevitable distortions that appear within any twodimensional photograph. “Is his nose really that big?,”she asks and we reply, “No, it’s pretty large, I

-46-

specifying an unfamiliar manifold through its charts

acknowledge, but this replacement photo supplies a better impression of itsmagnitude.” But, of course, this new snapshot will be itself marred by some furtherunwanted feature (certain theorems of map projection render these distortionsunavoidable), so we may need to cover Jeff’s nasal regions with yet a further chart. And we might also provide our friend with a metrical key that allows her to computeobjective nose length from its image in the photo. Adding photo about photo in thiscorrective manner, we abstract away the idiosyncratic features we don’t want toattribute to Uncle Jeff himself. Since we’ve not dealt with the back of his head yet,we must supply a further flurry of charts for its delineation (for the same reasons asthe earth requires a connected atlas of maps). But if we eventually surround ouruncle with a completed atlas of every photo that might be possibly taken of him(complete with transition maps telling us how to collate the information suppliedbetween charts), then we will have removed every occasion for erroneousconclusion on the part of our friend. But a full atlas of the desired stripe will requirean infinity of individual photographs.

In just this way, so mathematicians introduce their various flavors of“manifold”44 as the often unfamiliar invariant object” that lie behind the surroundingmaze of charts through which we become acquainted with them. So the standarddefinition of a “vector equation on a differentiable manifold” assembles a rich set ofcovering charts into an atlas that can cover every sector of the target manifold withsome local map or other, along with transition maps that tell us how to translate datafrom one overlapping chart to another. FIG: SPECIFYING AN UNFAMILIAR

MANIFOLD THROUGH ITS CHARTS Because each covering patch will typicallyexhibit features (such as an Euclidean metric)that shouldn’t be attributed to the targetmanifold, we remove this unwanted structure bypuffing up a basic set of covering charts toinclude every form of equally allowableprojection in the “photo album” manner justdescribed and mimicking the effects of non-flatdistances employing appropriate correctionrules.

Techniques of this sort, in which thenotion of abstracting an invariant, become precisified through the set theoreticconstruction of forming an equivalence class of charts, were pioneered by RichardDedekind in the late nineteenth century and have now become the standard means

-47-

for introducing novel “objects” into mathematics’ dominions. Recalling our Essay 5discussion of the blind men and their “understanding” of their elephant, the thinkingemployed rests upon the pragmatic assumption that we possess an adequateconception of a novel object or property once we know how to calculate suitablywithin its vicinity. Subtle variations upon this methodology, involving carefulconsideration of cooperative family harmonies, lay at the center of the semanticrepairs studied in Essay 8. All of these innovations allow us to maintain the spirit ofEulerian optimism through invoking mildly transcendental departures from rawcomputational capacity of a more sophisticated cast than merely “serving as thelimits of simple coordinate plottings.”

The central role of set-theoretic thinking within constructions of this ilkstrikes me as entirely natural. We confront a descriptive problem–a goose flyingacross the globe–that presents us lowly numerical calculators with a complex set ofcomputational opportunities and failures. Opportunity in the sense that overshortish spans an Euler’s rule calculation can parallel the bird’s flight to areasonable degree of accuracy (which we can always improve at the cost of greatercomputational labor). Failure in the sense that we will eventually obtain rubbishyresults if we don’t shift the basis of our computations swiftly enough. The usualdefinitions attendant upon “vector equation on a manifold” supply an excellentportrait of how these mildly transcendental relationships between target reality andconcrete computational capacity play out, allowing us to occupy an intellectualposition where we justly feel that we understand nature’s own processes quiteadequately (with respect to geese, at least), despite the fact that we can’t directlycalculate everything we might want to know about these behaviors. Such is theimproved form of Eulerian optimism to which Hertz and his compatriots would havehappily subscribed (I believe).

In assembling suitable informational packages (or references) for words like“manifold” (or “distribution”–Essay 8) in this pragmatic manner, I believe thatmathematicians have acted as subtler philosophers of language than many of us“professional” practitioners of the art. In so doing, they have provided us withricher models for understanding how concrete linguistic expressions can adequatelyreflect their physical targets, while simultaneously codifying the inferentialprocedures we should follow in reasoning about such circumstances successfully.45

As remarked earlier, many philosophers appear to have lost sight of thesebasic reasons for set theory’s vital intercessions within the worlds of appliedmathematics and instead regard the discipline’s key objectives as largely“reductive” (= reducing ontological commitments) or “mereological” (= capturing a

-48-

a non-“mathematical” curve

notion of “part” and “whole”). Focus on tasks of the latter cast encourages thefaulty impression that standard Zermelo-Fraenkel postulation overreaches from anaturalistic point of view. But this narrowed emphasis ignores the central role thatconsiderations of limits and algorithms, how structures unfold or collapse intosingularities, etc. must play in our scientific efforts to accurately gauge ourcomputational place within nature.

(x)

Thus far I have concentrated upon the happy offices that set theoreticcorrectives supply in allowing us to develop a portrait of our descriptive abilities inwhich an unblemished Eulerian optimism with respect to mathematics maintainspride of place. But dark clouds yet threaten this sunny faith. To appreciate theirnature, it will help to return to a second source of Cartesian descriptive pessimism,often dismissed as another artifact of his inadequate mathematical repertory, butwhich is suggestive of deeper concerns that we should bear in mind before werhapsodize too effusively with respect to our reformed optimism. Specifically,Descartes worried that curved paths can readily appear in nature (e.g., the randomconfiguration that a string assumes when casually tossed upon a table) which willnever submit to any form of precise description with which mathematics can capablydeal.

Geometry should not include lines that are like strings, in that they aresometimes straight and sometimes curved, since the ratios between straightand curved lines are not known, and I believe cannot be discovered byhuman minds, and therefore no conclusion based upon such ratios can beaccepted as rigorous and exact.46

He labels these “beyond the reach of geometricaldescription” curves as “mechanical” or “imaginary”(on the grounds that their contours can only berepresented as images within the faculty of theimagination and not through rules cognizable by ourpurely intellectual powers). FIG: A NON-MATHEMATICAL CURVE He cites the spiralsdrawn by the two-handled contraption illustrated as aparadigm of a “non-mathematical” curve:

Probably the real explanation of the refusal ofancient geometers to accept curves more complex than the conic sections

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lies in the fact that the first curves to which their attention was attractedhappened to be the spiral, the quadratrix, and similar curves, which reallydo belong only to mechanics, and are not among the curves that I thinkshould be included here, since they must be conceived of as described by twoseparate movements whose relation does not admit of exact determination.47

In other words, although Greek mathematicians mistakenly rejected many curvesaccepted as “geometrical” within Descartes’ own algebraic research program, theydid not err in withholding this honorific from our spiral curves.48 Why did hebelieve this? In modern terms, because he thought that any rule susceptible toprecise mathematical investigation could only contain a single independent variableas generator:

All points of those curves which we may call "geometric"...must bear adefinite relation to all points of a straight line and this relation must beexpressed by means of a single equation.49

Indeed, it took a considerable degree of mathematical advance before the notion oftwo or more independent variables came into view: the development of partialdifferential equations was greatly impeded thereby (although they represent thenatural vehicle for registering the physics of extended flexible bodies).50

Well, a simple Eulerian faith in differential equations can resolve theseparticular exclusions but Descartes’ underlying concerns still arise in moresophisticated ways, although this fact was not widely recognized until JacquesHadamard’s penetrating critique of the early twentieth century.51 Suppose that wepour ink onto a table top from two independent bottles and assume that the flow isgoverned by suitable fluid equations. Under plausible physical assumptions, theWeierstrass approximation theorem declares that these puddles can be approximatedas closely as we like by so-called analytic functions (most garden variety calculationrules are of this class), leading Henri Poincaré to rashly declare:

The physicist may, therefore, at will suppose that the function studied iscontinuous, or that it is discontinuous; that it has or has not a derivative;and may do so without fear of ever being contradicted, either by presentexperience or any future experiment. We see that with such liberty he makessport of difficulties that stop the analyst.52

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the descriptive excesses of analytic functions

But this jaunty point of view leads to a very odd result: if we model our pools of inkby an analytic function, then the properties of the right hand puddle can becalculated completely on the basis of the properties encountered on the left, despitethe fact that they may have been poured from entirely independent bottles.Articulated in terms of a numerical simulation, this connective linkage allows us fillin the graph paper under the right hand bottle as soon as we know how to fill in thegraph under the one on the left. FIG: THE DESCRIPTIVE EXCESSES OFDIFFERENTIAL EQUATIONS Why? Because analytic functions inherentlyembody a strong form of “rule” (called “analytic continuation”) that forever bindsits pieces together, in an excessively tight manner reminiscent of the “untrue tonature” character embodied within Descartes’ restricted class of “single generating

variable” functions. Hadamard concludes that an apt modeling of independent inkpourings requires that we employ a wider class of functions that admit non-analyticbreaks of some sort or other. It is only after the input flows have halted and the inkhas mixed into a final rest distribution that the connected character of an analyticfunction becomes appropriate as a modeling device (because all portions of theconjoined puddle have now had time to communicate with one anotherthoroughly).53

Here’s a second striking illustration of the flaws of analytic function over-description involving the steady flow of liquid from a spigot in a beer keg. Due tothe steady state circumstances, it is not surprising that the governing equation(Poisson’s familiar 2z/x2 + 2z/2y2 = ρ) accepts analytic solutions. Yet look at

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nozzle behaviors

problem of connecting data across a tear

the diagram on the right: that is ourequation’s uninterrupted “true solution.” Yet every serious patron of the barroomrecognizes that beer does not gush forth inthat way. FIG: NOZZLE BEHAVIORS Substituting “tea” for “beer,” A.I. Rubamand J.S.B. Gajjar write:

A [naive] “mathematician” couldexpect that the trajectories of the fluidparticles in the jet should exactly coincide with thelines in [the right hand] figure [given that they satisfythe same modeling equation and side conditions]. However, an “experimentalist” and, in fact, anyonewho has observed how tea is served , would disagree... The fluid is never observedto turn around the edges of the [nozzle walls] ... This dilemma led Helmholtz to aconjecture that, in addition to the smooth solution shown [on the right], the[relevant] equation also allows for a solution [displayed on the left] where the fluidvelocity has a jump across the boundaries of the jet.54

In short, Helmholtz tacitly liberates his newly tolerated slip flows (as they areconventionally labeled) from prior obligations to fully obey their governingequations along these jet interfaces (this is but the first of a long list of specialexceptions that gradually creep into applied mathematical practice). It is oftensurprising that it often takes a long time before the need for these special exceptionsgets noticed–Helmholtz wrote a full one hundred years after Euler). However,tolerating these special exceptions creates immediate problems of data harmony ofthe sort we investigated in other essays. Let’s look at the somewhat simplerproblem of computing the interior tensions in a drumhead tightened in a prescribednumerical manner around its periphery. Our basic computational task is to fill in theinterior squares based upon these exterior tensions. As observed in Essay 2, wegenerally attempt to build up a graphical solution in the manner of a crosswordpuzzle: we start by constructing partial solutions employing the five point stencilrules we can extract from our modeling equation and then tinker with theseincomplete proposals until they properly match up everywhere in the interior (i.e.,they satisfy our stencil rules along their joins as well). However, if we wantonlyintroduce a tear into the drum membrane and say nothing more about how thesquares on each side of the tear should relate to one another (we’ve turned off theapplicability of our stencil rules along the rip but not offered any replacements), then

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Hadamard

we plainly won’t be able to complete our graph paper crossword puzzle (or, moreexactly, we can fill it out in virtually any manner we choose). To fill the void leftby the inapplicability of our stencil rule, we must seek some plausible connectionformula applicable within the ripped region (a common repair: adjacent squaresexert no normal pull whatsoever across the tear). FIG: PROBLEM OFCONNECTING DATA ACROSS A TEAR As soon as we do this, non-trivialquestions of data harmonization immediately arise: will our suggested repair rulesallow us to fill out our crossword puzzle in a unique manner or does some moredelicate adjustment need to be found, in the manner of Essay 8?

Formally, the repair rules we seek assume the form (at the original differentialequation level) of “matching conditions across an interface” (e.g., our tear or theHelmholtz’ slip condition allowances). As such, they present us with a seriousdescriptive dilemma emerges as central in many of our other essays: the fact that thephysical processes active in the regions where we opportunistically locate ourinterface condition allowances tend to be quite elaborate yet, mathematically, thedemands of data harmony repair require us to model these same complex regions incrude and, generally, lower dimensional terms (in Essay 2, I call this thephenomenon of unequal data registration). Such considerationslead us to the realization that we can’t regard the interiordifferential equations we employ within a modeling as codifyingall of the pertinent physics, for such descriptions invariablyoperate in cooperative harmony with side condition data such asboundary and interfacial conditions in which different forms ofphysical data are also registered, albeit in a more compressedmanner. Long ago (1911) Jacques Hadamard remarked onthese cooperative entanglements as follows:

Each partial differential equation gives rise, therefore,not to one general problem, consisting in theinvestigation of all solutions altogether, but to a numberof definite problems, each of them consisting in the research of one peculiarsolution, defined, not be the differential equation alone, but by the system ofthat equation and some accessory data.55

He also observed that simpler ODE modelings (such as employed in celestialmechanics) do not present the same difficulties of data harmonization, a sage remarkto which we will later return. FIG: HADAMARD

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Suppressing compression in the Wren-Huygens treatment

Through completely ignoring Hadamard’s cogent warnings, I find it rathermaddening that Theory T enthusiasts will prattle merrily about the explanatoryvirtues of “laws”56 without a proper acknowledgment of the significant descriptiverole played by Hadamard’s “accessory data.” But the future prospects fordescriptive optimism of an improved Eulerian cast depend critically upon ourabilities to counteract the effects of unequal data registration. To see this clearly, itwill help to review the role that asymptotic matching played in motivating thelimiting opportunisms of earlier authors. We have already observed Descartes’concerns about our mathematical abilities to track the complex flows inside aconstricting pipe: he instead pointed out that we can work usually work around thisstretch of “indefiniteness” matching upstream and downstream flows through acanonical form of matching principle involving (what we’d now call) energyconservation. In an allied vein, Leibniz pointed out that popular approaches to theelastic collisions of billiard balls in the simple manner articulated by Huygens andWren (and developed by Newton) proceeds as if the colliding spheroids neverdistort at any stage in their encounter. But this gambit can only be regarded as aconvenient mathematical trick, for the direct observation of slow elastic collisions inreal life (e.g., between beach balls) readily reveals that the balls compress andreexpand when they collide and do not instantaneously rebound without alterationof shape as presumed in the Wren-Huygens treatment (Leibniz cites Mariotte’sobservations on this score). FIG: SUPPRESSING COMPRESSIONS In otherwords, their scheme effectively cuts off all consideration of the complicated internalevents that transpire on atime scale swifter than Δt*and asymptoticallyconnects up ingoing andoutgoing motions on thetwo sides of the collisionthrough conservation lawsin the stock manner ofevery freshman physicstext (often utilizing Newton’s coarse coefficient of restitution parameter as asupplementary fudge factor). Methodologically, we are skipping past the collisionevents within Δt* in exactly the same manner that Cartesians evaded the “indefiniteprocesses” that reshape particle flow within a constricted pipe. This treatment isasymptotic in the sense discussed earlier: it artificially collapses the short interval oftime Δt* in which the balls actively contact one another down to a event singularity

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Leibniz

of extent 0.To understand the collision process rightly (and to gauge the ensuing physical

effects with greater accuracy), Leibniz observed that we need to open up thecollapsed Δt* interval and investigate the sundry ball compressions and expansionsthat transpire inside. This “inside Δt*” approach demands that considerably revisethe physical ontology with which we operate and now model our balls as flexibleelastic bodies, capable of temporally storing energy in the form of strain, leading inturn to a very complicated moving boundary value problem that could not beeffectively addressed by numerical methods until far into the twentieth century (eventoday, most practical calculations of collision scattering rely upon asymptoticalternatives). If any of the early pioneers of early modern science had insisted uponfollowing a refined treatment, the development of “billiard ball physics” would havebeen halted in its infancy.

A mathematical optimist of an Eulerian stripe will maintain that only thelower scale, flexible ball model describes the “true physics” of the processcorrectly, but Leibniz himself maintained a more pessimisticthesis that we might label the improvable, yet unavoidable,need for asymptotic cutoff appeals within science. FIG:LEIBNIZ Specifically, he believed that, although we couldclearly improve our billiard ball descriptions through theinvocation of the fancier species of mathematics inherent incontinuum mechanical thinking, even the latter will not be ableto evade subtler cutoff policies of its own. In other words, anymode of scientific description must opportunistically paperover various descriptive complexities, leaving futureimprovements in these cut off arenas to some further compromised scheme. Hewrote:

Accordingly, if we think of bodies only under mathematical concepts likesize, shape, place and their modification and introduce the modification ofvelocity only at the instant of collision, without resorting to metaphysicalconcepts, i.e., therefore, without going into what form has to do with activeforce and matter with passive force--in other words, if we must determine thedata of collision only through geometrical configurations of the velocities,the result will follow, as I have shown, that the velocity of the smaller bodywill be imparted to a much bigger body that it meets.57

As it happens, his particular reasons for presuming that opaque forms of asymptoticpatching together represent an ineliminable, though moveable, aspect of

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mathematized physics not longer seem pertinent58, but allied theses are quitecommon in the development of science. For example, I believe that Niels Bohr’smurky remarks on the symbiotic relationships that quantum and classical modes of description bear to one another (which is one of the several relationships that hecalled “complementarity”) are best interpreted in this vein. The manner in whichcurrent science ties together the levels within a complex material throughhomogenization techniques raises these basic concerns in a very serious manner, asEssay 5 documents.

Leibniz likewise realized that most differential equation models areconstructed by artificially scaling the upper scale regularities in a material (such asthe phenolic resin of modern billiard balls) down to the infinitesimal level at whichdifferential equations operate. In doing so, we paper over the plain fact thatobservation under a microscopic reveals the tangled polymers of which it iscomposed, which do not stretch or twist in the same simple manner as larger hunksof the plastic. But if we refuse to extend our upper scale modeling down to thislowly level, we deprive ourselves of a large amount of “analytic data” about our ballthat only emerges when we consult the solutions (or partial solutions) of ourperfected differential equation (this phenomenon, further discussed in Essay 6,demonstrates the informational virtues of the perfect volcanos located in the generalstrategic neighborhood of the less perfected real mountains). This basicconsideration reinforces the fact that we should rarely regard differential equationsas standalone instruments of the highest descriptive caliber, in contrast to their moreimpoverished boundary and interfacial companions. Rather we should approach theensemble as working together as a cooperative family.

Essay 8 reviews in some detail the efforts that modern mathematicians havedevoted towards correcting descriptive imbalances of a “differential equations do itall” character. As stated before, in working out these subtler forms of dataregistration, these mathematicians have considerably expanded our understanding ofthe subtle manners in which a linguistic formalism can capture valuable informationabout a target system without requiring a simple encoding of a “is a bird/ belongingto class aves” character. Such conceptual improvements strike me as promoting thecause of a subtler descriptive optimism in a successful way, but many other forms ofasymptotic stitching have so far resisted comparable elucidation insofar as I cansee.59

And this brings us to our deepest reasons for opposing the static conceptionof mathematics’s obligations within science promoted by Quine and others: the falsenotion that science’s needs for mathematics can be neatly codified within some

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governing set of posits, whose obligations an enterprising philosopher might attemptto whittle down through weird forms of syntactic paraphrase. Epistemologically,we instead find ourselves confronted with a chapter in the natural history of homosapiens that we don’t know how to fully write at present. We can’t presently gaugethe degree to which future improvements in mathematical thinking will allow usimprove our descriptive grip on nature in a manner that transcends the necessities ofjumping brutely from one descriptive opportunity to another. As we’ve seen, infashioning its surprising conceptual remedies mathematicians characteristically drawupon every corner of Greater Mathematicsland for helpful hints and, as long as weremain ignorant of our full capacities in these respects, the naturalistic offices thatmathematical thinking performs on our behalf can never be consigned to anyrestricted group of tidy posits. Lacking the powers of occult prognostication, I entertain no firm convictionson what future science will eventually look like. But neither should the scores ofanalytic metaphysicians and philosophers of science who write as if they knewotherwise. Surely if philosophy is to adequately explore the “range of possibilities”open before us, it should not close the book prematurely on well-motivated doubtswith respect to the capacities of mathematical representation. I think that unbridledspeculation on the scientific hereafter is best left to writers of science fiction andphilosophers of the cracker barrel school. In the absence of futuristic assurance,academic philosophers should actively explore the possibilities of alternative modesof scientific development and not insist upon a debatable future simply because itmakes our philosophizing easier.

Let me close this section by returning to Hadamard’s observation that thetroublesome demands of cooperative family coordination between distinct modelingingredients become considerably greater within the realm of PDEs (=PartialDifferential Equations and their more sophisticated cousins) than amongst thesimpler ODEs (= Ordinary Differential Equations) employed within Newtoniancelestial mechanics and allied applications. One of Hadamard’s great contributionsto mathematical understandings lies in his recognition of how differently ourexplanatory landscapes diversify and alter under this simple change in formalism(much of the argument of my entire book is contained within this pithy remark). Historically, Hadamard’s observations were required to stop mathematicians from“looking at PDEs through ODE eyes.” It strikes me that the logic-biaseddistortions of Theory T thinking (e.g., neglect of equational signature, misuse of“boundary condition,” etc.) trace to the same mistaken propensities for collapsingsubtle forms of mathematical modelings onto simplistic ODE-like contours that they

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two fearsome books

fail to fit. The descriptive glories that Newton once achieved through workingwithin a simple ODE framework can no longer be simply continued; if we wish toremain prophets of mathematical optimism with respect to the physical world, wemust struggle more intently with the real life problems of ratifying its demandingexpectations.

(xi)

The story just sketched strikes me as intrinsically “naturalistic” in anyplausible sense that I can supply for this misty term, for it charts a roughlyevolutionary sketch of the intellectual tools that humans employ to elevate theirdescriptive capacities beyond the rude hunter-gatherer level with which we began(and to which we remain forever yoked). As noted in our introductory section,

some misbegotten conception of scientisticrequirement has unleashed an army of reformerswho maintain that naturalism’s honor can beredeemed only through odd programs for rewritingthe contents of our scientific textbooks in peculiarways. What strange intoxicant has sent them alongthis path? Much of it traces to the two scripturesmentioned at the outset: Quine’s From a LogicalPoint of View and Paul Benacerraf’s “MathematicalTruth.” FIG: TWO FEARSOME BOOKS

It is easy to discern where Quine goes astray–it stems from his presumptionthat it is profitable to think of science as operating in spurts of all-at-oncepostulation: at time t, we posit an all-embracing Theory T which supplies us with thefixed vocabularies we should employ while operating within T’s ambit. When T’sempirical fortunes turn against it, we seek a replacement T’ and proceed as before. This is a view of science from whose ledgers the ongoing necessities of strategicadaptation, innovation and monitoring have been thoroughly scrubbed and quicklyleads to the faulty dictum that the only mathematics that physics requires for its ownpurposes is whatever expressive tools are required to express its fundamentalpostulates (a presumption that immediately reduces physics’ “needs” to some lowlylevel within set theory’s analytical hierarchy). As we’ve observed, this

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insight pops up

characterization has the true interdependencies reversed:physics must continuallylook to general mathematical thinking for novel strategicsuggestion and for arenas in which clotted reasoningpolicies can be unraveled adequately enough that theirmerits can be reliably adjudicated. Insofar as I candetermine, potent advice on either of these scores mightpotentially pop up, Whac-a-Mole fashion, within anyquarter of mathematics’ far reaching dominions. FIG:INSIGHT POPS UP

Nevertheless, at core Quine was a pragmatic empiricist with respect to“linguistic meaning” (just as I am) and might have cheerfully accepted many of themethodological correctives suggested here (all-at-once postulation does not accordhappily with his Neurath’s boat proclivities). But Paul Benacerraf’s owncontribution to naturalistic ensnarement traces to a more traditional thesis aboutlanguage that Quine would have rejected. It might be dubbed the semantic rigidity ofphysical vocabulary.

This doctrine maintains that, with respect to physical terminology (but notmathematics), we possess a firm and constant conception of the referential facts thatmust obtain within the external world for the sentential applications of thatvocabulary to qualify as true or false. Benacerraf writes:

[A physical] proposition p places restrictions on what the world can be like. Our knowledge of the world, combined with our understanding of therestrictions placed by p, given by the [referential] truth-conditions of p, willoften tell us that a given individual [could or] could not have come intopossession of evidence sufficient to come to know that p.60

He further claims that, courtesy of Alfred Tarski, we enjoy a firm grip on how thisunderstanding of truth-conditions structurally relates to the inferential policies weshould apply to “propositions p”:

My bias for what I call a Tarskian theory [of truth] stems simply from thefact that he has given us the only viable systematic general account we haveof truth. So, one consequence of the economy attending th[is] standard viewis that logical relations are subject to uniform treatment: they are invariantwith subject matter. Indeed, they help define the concept of “subject matter. The same rules of inference may be used and their use accounted for by thesame theory which provides us with our ordinary account of inference, thusavoiding a double standard.61

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don’t trust this man!

Without recognizing that he has done so, Benacceraf in one fell swoop hasobliterated all of the naturalistic tasks for mathematics that have concerned us withinthis essay. All gainful employment removed from view, mathematics assumes theunsettling aspect of a Zachary Scott-style lounger whose motives and objectivesappear suspicious and unnaturalistic. FIG: DON’T TRUST THIS MAN! Why? Because our physical claims experience no difficulties in aligning themselves withexterior truth-values while those of mathematics cannot tie down their ownreferences in the same direct way. In consequence, its sundry posits appear weirdlydisconnected from the causal bonds that attach our physicalvocabularies to the world.

Indeed, from such a vantage point we should worry: werecognize how Mildred Pierce earns her money, but MonteBeragon’s sources elude us.

No pragmatist worth their salt should accept the premise thatour everyday physical vocabulary earns its inferential andreferential credentials in this direct and easy manner, but I will notlitigate that case here. Whatever the situation might be with wordsof popular discourse, Benacerraf’s claims surely cannot apply to the physicalterminologies of which science is composed and his claims otherwise trade upon thecarelessness with which he invokes the term “truth-condition.” As other essays inthis collection observe in greater detail (e.g., Essays 8 and 9), we often findourselves obliged to scrutinize critically the supportive semantics of establishedterminologies that were previously regarded as well understood. Consider this run-of-mill characterization of Isaac Newton’s celebrated discoveries about the“composition” of light:

In 1666, Sir Isaac Newton did a famous experiment that revealed therelationship between light and color. A beam of white light, in his case,sunlight, was passed through a triangular prism. The beam was diverted toa different direction and at the same time dispersed into a spectrum. ... Heobserved that [the hues produced] were components of white light. He thenpassed the colored spectrum light through a prism that deviated the lightback in the original direction. When he did so, the components of the lightrecombined to make white light. Thus he showed that white light could beseparated into component colors and that the component colors could berecombined into white light.62

We can fairly evaluate this assessment as “correct” insofar as it goes (although wemight legitimately criticize it for “not going far enough”63). What should we say

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about the “truth-conditions” of the terms “component” and “composition” employedthroughout? Here is the answer I would supply. Newton’s discoveries justifydescriptively employing a strategy of factoring light’s behavior into “components”and the profitable reasoning policies that stem from this realization naturallyentangle the word “component” with real properties of light in a reference-like way. These empirical facts provide the word “component” with a clear enough worldlyreading that most of the claims made in the passage quoted come out as “true.” ButNewton himself (and most of the rest of us unless we have learned about the strangevicissitudes of light’s career documented in modern books) suffered from anincorrect picture of how this word/world linkage really transpires. This is becausethe phase relationships integral to attributions of “color” (or wavelength) withinordinary light unexpectedly need to be understood in the elaborate stochasticmanner that we applied to our jet of beer earlier. Appropriate interpretations of“component” and “factor” need to follow suit as well. However, Newton’s ownfaulty (although fully pardonable!) picture suggests that “x is a beam of white light”should be “semantically analyzable” into conjunctive components “ x is composedof n1% red light & x is composed of n2% yellow light & x is composed of n3% greenlight & ....” , where our list runs through all of the spectral colors and the ni’s sumto 1. This conjunctive analysis erroneously suggests that further inferentialconclusions with respect to our light are logically warranted but many of these willprove incorrect when tested (example: the conjunctive picture improperly suggeststhat we can meaningfully consider “light’s red component” on a very short timescale, a claim that must be rejected under a proper stochastic analysis). As such,these unexpected blockages of “logical” reasoning rules stem from the sameadjustments in practical inference that doom the early apriori certitudes discussed inEssay 6. I won’t rehearse those issues again here.

Benacerraf appeals to Tarski’s approach to “truth” and “linguistic meaning”as a means of segregating “statements resting upon clear referential piers” (thephysical claims of science) from those that don’t (the typical claims of puremathematics). But this assumption rests upon an excessively rigidifiedunderstanding of what Tarski’s researches about “semantics” tell us about reasoningwithin a language subject to the normal pressures of improving development. FIG:TARSKI As Essay 1 emphasizes, his soundness proofs with respect to (what lookslike) logical reasoning carry probative merit only when the vocabulary studiedrelates to its external world supports in the simple predicate to extension picture heposits (i.e., that “is a bird” picks out a simple corresponding set {x| x is a member ofthe class aves}). Well, sometimes these simple semantic pictures hold up as

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Tarski

opportunity overridinglogical pattern

appropriate upon further study but sometimes they don’t. Our original simplisticassessment of what a “component of light” represents, met the latter fate. Accordingly, the inferential warrants provided through Tarksi-style soundness

proofs are always provisional, hostage to the consideration that theymay rest upon a faulty picture of how physical information isactually encoded within a descriptive language. Sound reasoningwith respect to Newton’s decompositional discoveries de facto restsupon subtler forms of stochastic factoring than Newton, or anyoneelse, could have anticipated at the time. And all of appliedmathematics’ allied forms of “correctness proof” provisionally restupon word/world pictures in the same defeasible way. We have

excellent results that establish important error bounds for application of Euler’s rulereasoning with respect to differential equations, but they don’t count for much whenthe target “differential equation” doesn’t store data in the manner we havepresumed.64 That does not mean that the insights supplied by these proofs are notvaluable, but merely that conclusions of considerable “value” sometimes mislead.

Does all of this show that “logic is empirical”? No, but it merely shows thatthe semantic pictures (= theses about word/world relationships) upon which we relyin deciding whether our vocabularies conform to familiar logical categories or notsometimes prove erroneous. Just so: the Newtonian picture of the physicalunderpinnings of the word “component” encourages the further assumption that the“&”’s in Newton’s “conjunctive analyses” are supported by worldly fact in amanner that renders a Tarskian “soundness proof” relevant to Newton’s furtherreasonings. But should Newton bet the farm without cavil upon the assumption thatany detachment and modus ponens operations he applies to these “conjunctions”will forever unfold without a blemish? No; the upright pathways of pure logicsometimes deposit us in horrible coal pits. 65

But the problem lies with erroneous semantic picture, ratherthan logic itself.

I’ve sometimes found that the following metaphor helpsclarify the situation. Logic lays out the inferential boulevardsof a scientific discipline in tidy, Midwestern order. Unfortunately, a cruel giant (= real world application)presently looms over the town and will stomp on anyone whoventures into his shadow. FIG: OPPORTUNITYOVERRIDING LOGICAL PATTERN Moral: stay out of theshadow region even if a soundness proof apparently

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guarantees that a trustworthy road runs through it. And the same limitations inauthority pertain to the standard correctness proofs for numerical methods, for thecorrelational portraits upon which their applications rest often prove irrelevant to thecircumstances to which they will be concretely employed (reappraisals of this stripehave proved a commonplace occurrence within the annals of computing).

Philosophers of a Theory T persuasion commonly presume that all ofscience’s vast menagerie of “useful reasoning practices” can be neatly codified as“instances of logical reasoning from clearly enunciated premises.”66 A pragmatistthinker such as Quine will subscribe to such a thesis because he believes thatimproving “Science”67 invariably obeys a basic methodological obligation toamalgamate all of its findings into one huge, axiomatically organized unity ofscience glob. But as our other essays have stressed, real life scientific progressoften unfolds in a contrary manner, in which effective modeling doctrine oftencondenses into isolated patches (or “protectorates”) held together throughhomogenization and other forms of asymptotic stitching. And this is because theinferential imperatives pertaining to correctly implementing a strategic opportunityinvolving factoring (such as Newton uncovered) generally far outweigh the utilitiesof following otherwise unmotivated strands of logical reasoning (the imperatives offactoring often serve as the overpowering giant in the metaphor above). Quine’sexaggerated forms of semantic rigidity derive from these faulty methodologicalpresumptions, whereas Benacceraf’s variant traces instead to mistaken assumptionsthat our initial pictures of word/world relationship invariably maintain themselvesover time. Either way, such views cast significant obstacles in the pathway of thenaturalist pilgrim and should be cast aside.

With respect to this collection’s broader themes, we should not seek alanguage that irrevocably binds its physical terms to the world in the rigid mannersof either Quine or Benacerraf, for such semantic carpentry, were it possible at all,would not construct a linguistic domicile suited to human requirements, which mustforever adapt and diversify in response to nature’s whimsically shifting currents. Instead, we must steer our usages by profiting from whatever fresh data we canacquire with respect to the strategies that underpin successful reasoning across awide array of settings, in the hopes that one of these transferred variants mightprove adaptable to the descriptive challenges immediately before us. Only in thatway can we adequately recognize the central place of “mathematical thinking.”

(xii)

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Pilgrim’s escape from Doubting Castle

This essay has attempted to parallel Penelope Maddy’s journey through thewilds of naturalism, rather as Christina followed after Christian in the originalPilgrim’s Progress, confronting vexsome obstacles that we subdue only throughclever applications of unanticipated physics avoidance. At the end of our ownpilgrimage, we find that our pathway has conveyed us to a certain “indispensibilityof mathematics” thesis, but not in the Quinean guise that we must “posit” a certainstretch of mathematics for science’s “purposes” (whatever those are supposed tobe!). We have instead reached the humbler recognition that significant aspects ofour abilities to extend our descriptive grip on nature involve mathematical thinkingas a continuing, if fallible, part of our improving activities, evolving from the lowlycomputational capacities we inherit from our hunter/gatherer forebears. As part ofthese improvements, we have found that set theoretic language provides the naturalvocabulary for articulating the mildly transcendent manners in which our concretecomputational capacities appear to relate to nature’s own unfolding processes. Atthe same time, we have observed that the mists of uncharted word/worldrelationships still obscure many of our most profitable techniques of dataregistration (differential equation over-scaling, excessive “analytic function”connection, asymptotic communications between scales, etc.), prompting us towonder how ably we can continue to advance along our present lines of set-assistedimprovement and at what stage must we seek replacement diagnoses or sullenlyretreat to descriptive opportunism of a Cartesian ilk. We presently lack convincingmethods for resolving these issues, for they remain at root entirely empiricalquestions--it’s up to nature to decide, not us--and do not represent issues we canexpect to resolve definitively at our particular moment in developmental time. Presently, mathematical and physical language operate together in cooperativeenterprise, in which sorting out the referential contributions of component strandsoften remains a perplexing task. As a consequence, our own journey has not led usto the cleanly insulated practices that Maddy discerns,but to a more compromised kingdom in which we mustcontinually struggle with the strategic underpinnings oftricky words. Their operations cannot be unraveledthrough aprioristic philosophizing but demand thecontinual, revisable scrutiny characteristic of anyscientific investigation into the evolving ethology ofhuman conceptual development. And our greatest toolfor understanding the mysterious byways of effectivestrategy is mathematics, of a rather purist cast. FIG:

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1. The Pilgrim’s Progress (New York: Dover, 2003), p. 9.

2. W.V.O Quine, From a Logical Point of View (Cambridge: Harvard UniversityPress, 1980); Hilary Putnam and Paul Benacerraf, Philosophy of Mathematics:Selected Readings 2nd edition (Cambridge: Cambridge University Press; 1984) andPenelope Maddy, Second Philosophy (Oxford: Oxford University Press, 2007).

3. Such is the attitude that the phrase “second philosophy” conveys.

4. Pursuit of Truth (Cambridge: Harvard University Press, 1990), p. 95, and “Replyto Parsons” in Lewis Hahn and Paul Schlipp, eds, The Philosophy of W. V. Quine(La Salle: Open Court, 1986), p. 400. I’ve extracted these citations from anexcellent discussion in Penelope Maddy, Naturalism in Mathematics (Oxford:Oxford University Press, 1997), p. 106.

5. Two variants: Hartry Field, Science without Numbers (Princeton: PrincetonUniversity Press, 1980) and Stephen Yablo, “Go Figure: A Path ThroughFictionalism,” Midwest Studies in Philosophy, 25: 2002 72–102. These effortsremind me of Georges Perec who derived mental satisfaction from composing anovel that assiduously avoids the letter “e.”

6. My sensitivities on the subject of gurus trace to the experience of having livedthrough the Age of Aquarius and enduring hours of earnest drivel conveying noascertainable truth-values whatsoever. See WS for more on this theme.

7. Philosophical Grammar (Berkeley: University of California Press, 1977), AnthonyKenny, trans., p. 381.

8. This crossroads is easy to discern; it occurs on pages 194-5 of SecondPhilosophy, op cit. She wants to detach her endeavors from worries about linguisticreference, but I see mathematics as playing an important role in helping usunderstand how subtle referential strategies operate, in a manner to be developedlater in the essay. I should add that Maddy’s appeals to practices are not asabsolutist as characterized here; they merely serve to staunch the distortions of

PILGRIM’S ESCAPE FROM DOUBTING CASTLE

Endnotes:

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Quinean indispensability arguments through a direct report on the motivations of theset theorists who work on foundations. As such, her approach is largelycomplimentary to my own. In this essay, however, I attempt to correct Quine’smisapprehensions from within applied mathematics itself, by reaching a warmerappreciation of set theory’s positive roles within a developing science. And so mypilgrimage travels through different territories than her own.

9. (the proper merits of postulational axiomatization have been misunderstood aswell).

10. There are several different modes in which our mayfly can carry these off. SeeSteven Vogel, Life’s Devices (Princeton” Princeton University Press, 1988). Findpage.

11. Knut Schmidt-Nielsen, Scaling Why is Animal Size so Important? (Cambridge:Cambridge University Press, 1984), p. 9.

12. See my "Mechanism and Fracture in Cartesian Physics," Topoi 14, 1997. Icould improve on this discussion were I to rewrite it today, but I believe that thegeneral portrayal of fracture and fusion sketched there remains sound. For more on“indefiniteness,” see Alan Nelson, “Micro-chaos and Idealization in CartesianPhysics,” Philosophical Studies (1995).

13. Jacques Rohault, Rohault's System of Natural Philosophy, Vol. 1 translated withnotes by Samuel Clarke (Farmington: Gale Ecco, ND)(copy of1723 original), pp. 33and 37.

14. Allied forms of asymptotic connection appear in the Huygens-Wren treatment ofbilliard ball collision discussed in other essays.

15. I’ve tacitly assumed that we are working with an ODE model of a purelyevolutionary character and so no boundary conditions are required. Historically,this same ODE suppression of complexity plays a large role in encouragingunrealistic conceptions of mathematical optimism long past their proper expirationdate, as mathematicians and philosophers have often persisted in looking at PDEs“through ODE eyes.” I believe that a significant portion of Theory T appeal tracesto this source.

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16. Lord Kelvin, "Electrical Units of Measurement" in Popular Lectures andAddresses, vol. 1 (New York: Macmillan, 1891).

17. I am reminded of Osborne Reynolds’ perceptive comment in his celebrated , “On Vortex Motion”:

It would seem that a certain pride in mathematics has prevented thoseengaged in these investigations from availing themselves of methods whichmight reflect on the infallibility of reason.

(Papers on Mechanical and Physical Subjects, Vol.1 (Cambridge: CambridgeUniversity press, 1900), p. 185). Just so, a certain pride in the philosophicallyascertainable has lead to a neglect of considerations that might reflect upon theinfallibility of logic. The exclusive immersion within first-order logic characteristicof modern philosophical training often leaves its pupils with an inaccurate estimationof the inferential uncertainties and out-of-country detours that appliedmathematicians confront upon a daily basis. The deductive roads to reliable resultswithin science are not so straightforward as witnessed in Logic 101, for hard-to-capture nature “hath inclosed my ways with hewn stone, it hath made my pathscrooked.”

18. The original invention of differential equations was inspired by such techniques,which long predated Euler.

19. This is an inherently set-theoretic constructions, for reasons we will stress later.

20. This is not to say that, in special circumstances, we may not be able to obtaindescriptive data of a higher quality, as when we manage to find explicit solutions toour equations. Such “analytic data” is highly prized and we should try to exploit itas ably as we can, in the manner of the firm rock in which alpine explorers sink theirgrappling hooks.

21. I should indicate that I am employing “transcendental” in the mathematician’ssense, not the philosopher’s, despite the fact that I teach at a school where the latteris all the rage.

22. As noted in Essay 6, Galileo writes as an unjustified optimist in The Analyst, forit is hard to see how such claims can be justified without some appeal toinfinitesimal relationships.

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23. Corrective supplements to this claim will be required later on.

24. Poincaré’s celebrated investigations into celestial mechanics only deepened ourappreciation of the potential gulf between feasible calculation and the physicalreality we “comprehend” at the differential equation level. He stresses the fact thatquite often we can concretely compute the long range trajectories of very simplephysical equations only in the vicinity of certain special opportunities (such asperiodic points), leaving us with only a “qualitative” understanding of the chaotictangles lying in between.

25. Such writers employ the term “law” in a very undisciplined manner, but theyfrequently mean something like “well-posed differential equation model withsuitable side conditions,” for only such an entity will possess “models” in anythinglike the sense intended.

26. In light of the complicated sentiments expressed in Letters to a GermanPrincess, these views should not be ascribed to the historical Euler. As has oftenhappens with terminologies that incorporate “Euler,” I exploit his name because (1)it supplies a compact label and (2) because Euler is properly credited with regardingNewton’s F = ma as a schema upon which concrete ordinary differential equationmodels can be built, following the “Eulerian recipe” expounded in my “What isClassical Physics Anyway?” in R. Batterman, ed., The Oxford Handbook inPhilosophy of Physics (Oxford: Oxford University Press, 2012). Historians ofmathematics remark on the fact that, for a long time beyond Euler’s mid-eighteenthcentury era, mathematicians continued to look upon more complex forms ofmathematical representation (such as partial differential equations and theirfunctional analysis cousins) “through ODE eyes.” These propensities becomefirmly arrested within the penetrating writings of Jacques Hadamard, which is whyhe frequently appears in these pages. As just remarked, most philosophers writingtoday seem to approach the the descriptive capacities of applied mathematics withthe same differential-equations-get-nature-right-at-an-infinitesimal-level optimismthat might have been fully justified within a simple, ODE-centered era. Documenting this accusation is difficult, however, because Theory T thinkingobscures these ODE contours beneath further layers of logic-focused gauze. Or so Iopine. What should seem undeniable is that subsequent demands on accuraterepresentation have considerably clouded the simple, mildly transcendentaloptimism that a naive ODE framework offers. Accordingly, old issues allied to“opportunism” should be reopened as we philosophically reflect upon these

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complicating features. Indeed, my central mission in this volume is one ofencouraging this reexamination. In particular, I recommend that, following Kelvin’squoted advice, philosophers should attend carefully to the exact manners in whichour concrete computations relate to nature through intervening pieces of descriptivesyntax such as differential equations. In my own thinking, I regard this task as amatter of “keeping track of how the pathways back to numbers actually run.” I findit distressing that many philosophers of science write glibly of “tensors” and“Hilbert spaces” today without any evident awareness of the specific correctionsupon computation that these two constructions codify. This is the point I mean todrive home when we fly our goose over a curved manifold later in the essay.

27. E.g. The Panda’s Thumb (New York: Norton, 1992).

28. R.J. Walker, Algebraic Curves (New York: Dover, 1950), p. 204.

29. Strictly speaking, the phase curves illustrated belong to some more regularsystem such as a pendulum, through which the attractor for the rattleback snakesthrough in a complicated manner.

30. As a general observation, closely linked to the modeling considerations of Essay5, the traits that most directly govern interactions occurring upon a particular scalelength (e.g., between our animal, its geographical environment and its food supplies)are aptly characterized in the “dominant behavior” terms relative to that scaleselection, because finer grained interactions are usually less reliably controlled.

31. On Riemann's Theory of Algebraic Functions and Their Integrals (New York:Dover, 1963).

32. I supply a lengthy discussion of Heaviside’s remarkable circumstances inChapter 8 of WS.

33. Irritated by confident philosophical assurances that “mathematical artifacts”within science can be readily distinguished from the genuine “physical” article, Idecided to test this claim experimentally by consulting a standard primer oncomplex variable methods in two dimensional elasticity. Prima facie, lots of“artifacts” should appear here, because the technique employs potential functions ofa complex variable to evade the vectorial considerations one would otherwiseemploy in this subject. Prima facie, we might anticipate that the multi-valued“jumps” that often appear with these functions should qualify as sterling exemplars

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of unwanted descriptive “artifacts.” But no: in the situations I looked at (stressconcentrations in plates with multiple cutouts), these “jumps” had parkedthemselves over unexpected but important relationships between the variousboundary stresses in a manner that greatly enhances our understanding of theircircumstances.

Such “artifact”/”non-artifact” claims stem entirely from excessively rigidifiedsemantical thinking and often serve as a jumping off point for the strands of“naturalistic” enterprise that operate on the assumption that physics and certainrestricted parts of mathematics operate in clean but separate isomorphic parallelism. As I later remark, I firmly reject all of these presumptions.

34. I write “structural arenas” because the useful borrowing of strategy from onedomain to another depends largely upon formal resemblances between the twotopics (often quite recondite), rather than requiring any further commonality insubject matter. That is why the general science of strategy lies almost entirelywithin mathematics of a “purist” flavor.

35. Hilary Putnam, Meaning and the Moral Sciences (London: Routledge and KeganPaul, 1978), p. 20.

36. In truth, a substantial measure of the set theoretic assurance is required to quellour prima facie worries that such loose appeals might never stabilize upon acoherent mathematical object at all.

37. Hermann Lotze, Logic (Oxford: Oxford University Press, 1888) Vol I, pp. 8-9. Translated by Bernard Bosanquet.

38. To be sure, these sorts of diagramatic “proof” present their own inferentialpitfalls, forcing a retreat to more reliable pathways of reasoning as articulated withintraditional Euclidian axiomatics. Similar motivations drive mathematicians from anaive direct Eulerian faith in differential equations to the set theoretic constructionswe shall discuss later.

39. Principles of the Theory of Heat (Dordecht: Reidel, 1986), p. 390. Mach is heremotivated to defend the credentials of an abstract “thermomechanics” in the modeof the Duhem of Essay 3.

40. Our frog notably lacks any means of assessing the reliability of its fly catchingroutines, beyond starving to death for lack of food.

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41. Persi Diaconis and Ron Graham, Magical Mathematics: The Mathematical Ideasthat Animate Great Magic Tricks (Princeton: Princeton University Press, 2012), p.4.

25. Michael Hartley (http://www.dr-mikes-math-games-for-kids.com/magic-number-cards.html).

26. The search shortening capacities of probabilistic algorithms are well appreciatedin this regard.

27. Stephen Macknik and Susana Martinez-Conde, Sleights of Mind (New York:Henry Holt, 2010), pp. 114-5.

28. Ibid, p. 7.

29. Tacitly, we should reset our inquiry within a complexified setting, but I’ll ignorethese complications here.

30. Tristan Needham, Visual Complex Analysis (Oxford: Oxford University Press,1997), p. 64.

31. I set the zero gauge of the potential energy at the height of the cannon’s mouth.

32. Unfortunately, these guarantees are pretty feeble for Euler’s rule and onegenerally seeks alternative methods that converge more quickly with a larger stepsize.

33. Central to many of these concerns were the failures of the “ Dirichlet principle”as pointed out by Weierstrass.

34. Combating these misapprehensions is one of Maddy’s chief objectives in SecondPhilosophy in which I fully concur.

35. Entdeckungen über die Theorie des Klanges (1787).

36. Lokenath Debnath and Dambaru Bhatta, Integral Transforms and TheirApplications, Second Edition, p. 3.

37. Modern mathematicians view a Sturm-Liouville decomposition as a map from“position space” into a dualized “energy space” in which we obtain “new eyes” forrecognizing physical capacities of which we were previously ignorant, underscoring

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our previous lessons in the importance of finding the right setting for rendering a setof reasoning practices pellucid.

38. Hans Sagan, Boundary and Eigenvalue Problems in Mathematical Physics (NewYork: Dover, 1989), p. 155. In general, I’ve followed Sagan’s development in myadmittedly brisk presentation here. The tactics involved are closely allied to the“shooting method” techniques of Essay 2.

39. Anton Zetti, Sturm-Liouville Theory (Providence: American MathematicalSociety, 2005), p ix. An excellent historical survey is J. Lützen and A. Mingarelli,“Charles François Sturm and Differential Equations” in J-C, Pont, ed., CollectedWorks of Charles François Sturm (Basel: Birkhäuser, 2009).

40. Mathematically, we confront an “eigenvalue problem” Lu = λu, where L is thedifferential operator from our target equation and the “eigenfunction” λ ties the sizerescaling to the frequency clock that drives the oscillations.

41. In point of historical fact, Liouville was a bit hazy about what a “limit” shouldbe and this led to various wrong conclusions on his part. The subsequent to the“completeness of the reals” pursued by Dedekind and others located his procedureswithin a genuine set theoretic frame (this is why Sturm and Liouville’s work issometimes characterized as “ur-set theoretic” in character–it relies uponassumptions about contractive “fixed points” that demand set theoretic tools forclarification).

42. Logically, we are not introducing new predicates Px through conventional non-creative definition (i.e., Px ... x ... where P doesn’t appear in the matrix), but viaphraseology that contains definite descriptions: ....(ιx)(x is a Sturm-Liouvillefactor)... Introductions of the latter sort qualify as legitimate “definitionalextensions” only if the implied existence claim can be ratified beforehand. Philosophers who should know better are often careless about “definability” issueswithin the present context.

43. The Principles of Mechanics (New York: Dover, 1956), p. 2. Although manytheses still actively considered within contemporary analytic philosophy (e.g., theso-called “thesis of extensionality”) took their original roots within late nineteenthcentury concern over the subtle problems of liberalizing scientific methodologywithout abandoning all appropriate standards of rigor, few practitioners today recallthese genuine difficulties with any degree of accuracy (e.g., they neglect the

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importance of investigating non-standard “fruitful definitions” for familiarterminologies). Some of this forgetfulness arises from an intervening generationallayer (the so-called “logical empiricists”) who frequently distorted the originalconcerns of their philosopher/ scientist predecessors in substantive ways (this hascertainly occurred with the “thesis of extensionality, whose central motivationsoriginally lay in underwriting the Sturm-Liouville claims just outlined, rather thanrepresenting the weird “methodological imperative” it becomes in Quine’s hands). WS, Chapter 6, surveys these issues at much greater (possibly excessive) length.

44. Many specific flavors of “manifold” can be introduced in more direct ways butthe standard approach to the diaphanous “differential manifolds” (as captured in O.Veblen and J.H.C. Whitehead, “Foundations of Differential Geometry” in Bull.Amer. Math. Soc. Volume 39, Number 5 (1933)) follows our pattern. Even here,alternative routes are available, but the Veblen-Whitehead treatment most directlyillustrates the patterns of thinking about mathematical existence that I seek toillustrate. Such “pasting together” techniques are implicit in Riemann and wereexplicitly developed (with respect to “analytic manifolds”) by Hermann Weyl in1913.

45. Here’s another way to characterize the conceptual shifts I have in mind. UnlikeLeibniz, Euler readily accepted as “mathematical” the curves drawn by differentialequations, even if they possessed no prior mathematical credentials. In doing so, hetrusted the “mildly transcendental” differential equation itself as an adequate arbiterof mathematical existence. But various descriptive foibles of the sort discussed herelater pushed mathematicians towards set theoretic construction as the most reliable“court of final appeal” for resolving problematic existence claims.

46. The Geometry of Rene Descartes, D.E. Smith and M.L. Latham, trans. (NewYork: Dover, 1954), p. 91.

47. Geometry, op cit, p. 44.

48. He also called such curves “imaginary,” meaning that their shapes could bepictured within the imagination only and not describable by a purely intellectualrule. See Mary Domski, “The Intelligibility of Motion and Construction: Descartes’Early Mathematics and Metaphysics,1619-1637,” Studies in History and Philosophyof Science (2009) 40. Our “crank/rocker” comment suggests an interesting glossupon Cartesian “indefiniteness.” In the absence of disturbances supplied by freely

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willing agents, the Cartesian inorganic world will grind on in the manner of agigantic piece of connected clockwork. In such a setting, the two controls of ourlittle spiral drawing device will again become connected to one another in “onedegree of mobility” fashion, but this time the mechanical entanglement transpiresthrough the entire “system of the world” and can’t be isolated within any palpablecrank/rocker mechanism. In such circumstances, our finite minds can’t adequatelyrepresent the spiral motions to our intellectual facilities due to the infinitecomplexity of the transmissive machinery involved. Perhaps the “indefiniteness” inthe fluid flow case can be approached similarly: as the fluid passes, the machineryof the “system of the world” causes the walls of the enclosing pipe to fleximperceptibly in a manner that allows the fluid matter to rearrange its shapeswithout ever reducing to a truly infinitesimal “dust” in the process.

I stress that I am unaware of any textual evidence in favor of this gloss on“indefiniteness” but it strikes me as a suggestion very much in the Cartesian spirit.

49. Geometry, op cit, p. 48.

50. See S.B. Engelsman, Families of Curves and the Origins of Partial Differentiation(Dordrecht: North-Holland, 1984). The barriers trace to early unclarities in theunderstanding of differentials and explain why Leibniz was not able to advance to afull PDE modeling of a dynamically loaded elastic material, of the sort required inbilliard ball impact.

51. Jacques Hadamard, Lectures on Cauchy’s Problem (New York: Dover, ND). See my Wandering Significance (Oxford: Oxford University Press, 1976), Chapter 4for more on this theme.

52. Henri Poincaré, “Analysis and Physics” in The Value of Science (New York:Dover, 1958).

53. These distinctions relate in turn to the differences between “elliptic andhyperbolic modeling circumstances” discussed in the “Physics Avoidance” essay inmy Physics Avoidance, op cit.

54. Anatoly I. Ruban and Jitesh S.B. Gajjar, Fluid Dynamics (Oxford: OxfordUniversity Pres Four Lectures on Mathematics s, 2014), pp. 2-3. In this vein, although I earlier intimated that we might alleviate Descartes’ pipe constrictionproblem simply by employing “differential equations,” actual circumstances requireus to instead utilize so-called “counterflows” involving sheets of fluid sliding past

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one another in Helmholtz’s fashion (see Vladimir Shtern, Counterflows: ParadoxicalFluid Mechanics Phenomena (Cambridge: Cambridge University Press, 2012)). Thisalteration forces mathematicians into addressing the non-trivial question of whether“jump conditions” can be adequately formulated in a manner that permits uniquesolutions within these liberalized circumstances.

55. Four Lectures on Mathematics (New York: Columbia University Press, 1915). As Essay 7 explains, many philosophers, operating from a blithe Theory Tperspective, presume that their “physically possible worlds” can be identified withthe “global models” allegedly posited within science. Hadamard’s remarks indicatethat PDEs rarely supply useful “solutions” of this unconstrained sort.

56. On the occasions in which one can adequately pin down what these authors aretalking about, they often mean “interior differential equations (of hyperbolicsignature)” when they write of “laws.”

57. “Specimen Dynamicum” in Philip P. Wiener, ed. Leibniz Selections (New York,1951), p. 129.

58. Although they are very interesting!–see Essay 3 for more on this.

59. Another central example of mysterious data registration analogous tohomogenization can be found amongst divergent series. Regular forms of seriesexpansion frequently display an annoying inclination to converge at an extremelyslow rate, in which thousands upon thousands of terms must be summed before theresults remotely approach the target behavior. One can easily be led intohorrendous error thereby but, unfortunately, reliable estimates on rates ofconvergence are hard to obtain. A godsend to practical computing emerges from theunexpected quarter of divergent series, summations that patently become infinitelylarge in the long run, yet supply extremely accurate characterizations of a targetbehavior as long we only pay attention to the initial terms in the expansion. Onegets the impression that these expressions have somehow reorganized the data theycontain in a manner that captures the system’s dominant behaviors within its initialterms, but oscillates madly with increasing desperation in a vain attempt to catch upthe finer details later on (rather as the governor in an out-of-control steam enginehunts ineffectively for a stablized position).

But this impressionistic “explanation” of divergent series behavior isundoubtedly anthropomorphic: we can attribute neither “wishes” nor an

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understanding of “dominant behavior” to a dumb formal series expansion! Nonetheless, practical science relies heavily upon their good offices in stitchingtogether its descriptive fabric. But how or why these linguistic manipulations workas they do largely remains a mystery at present.

60.“Mathematical Truth” in Putnam and Benacerraf, op cit , p. 413. Why any right-minded “naturalist” would succumb to the temptations of this semantic rigiditythesis is a mystery to me, for the doctrine was precisely concocted by philosopherslike Frege who yearned for a priori weaponry that might remain steadfast through allof our algorithmic struggles with nature. But surely a “naturalist” should recognizethat our position within the celestial frame doesn’t permit such an exalted estimationof our capabilities. On the Theory T side of these issues, analytic metaphysicians ofthe stripe surveyed in Essay 5 will warmly assure us that someday science willsupply us with a perfected Theory T from which all concerns of wobbly referencewill be entirely expunged, but I fail to discern the philosophical percentages inbetting heavily upon a horse that won’t cross the finish line until long after we areall dead.

61. Ibid, p. 411.

62. R. Kimber, R.W. Greiner and J.C. Heidt, eds. Quality Management Handbook2nd edition (Boco Raton: CRC Press, 1997), pp. 290-1.

63. Geoffrey Brooker nicely encapsulates the pedagogical dilemma of an instructorin modern optics:

The author of this book used words with more than usual care in Chapter 9,in an attempt at giving a correct impression and excluding commonmisconceptions; but the need for that care was itself an indication thatsomething better was called for.

Modern Classical Optics (Oxford: Oxford University Press, 2002), p.219.

64. The weak solution revisions of Essay 8 indicate the instabilities to which thesesemantic assessments are liable.

65. See my "Can We Trust Logical Form?," Journal of Philosophy XCI (1994).

66. This assumption clearly lies behind vainglorious claims (vide the DavidArmstrong quotation in Essay 8) that an author “understands adequately howscience works” as long as he or she has taken an elementary course in first order

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logic.

67. I capitalize the word because in writings like Quine’s ordinary canons ofcommonsense appraisal get personified into demiurges of uncanny methodologicalobligation (of an Ockham’s razor cast).