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 withrags, standing in a certain place, with two books in her hand and a greatburden upon her back. I looked and saw her open those books and readtherein; and, as she read, she wept and trembled; and, not being able longerto contain, she broke out with a lamentable cry, saying, "What shall I do?"

--John Bunyan (with apologies)

(i)

The unhappy pilgrim I witnessed was Penelope Maddy and the two fearsomebooks she held were Quine’s From a Logical Point of View and Benacerraf andPutnam’s Philosophy of Mathematics, second edition.1 She was embarking on ajourney that she reported upon in her Second Philosophy of 2007. 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, along pathways that seem beguiling at first but whicheventually entrap the traveler in dank sloughs and unwholesomedens of ignominy. This misdirection transpires under the allegedimperatives of “naturalist philosophy,” which prima facie soundsas if it should represent a Very Good Thing but somehow leads itsadherents to mutilate mathematics in ridiculous ways. SecondPhilosophy can be read (or, at least, so I’ve read it) as a valiantattempt to reorient the compass of “naturalism” (or some facsimile thereof) along amore reasonable axis. She begins her explorations by vowing that she will not bedistracted from her mission by the usual cavils of philosophical skeptics and willstrive to approach all issues in the customary vein of an empirical scientist (that is theattitude that “second philosophy” conveys). Nor will she accept methodological

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reformers

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

Maddy is concerned that adherence to such constrictive percepts is apt to clipthe wings of mathematics’ most stirring flights of fancy, 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 (he regards “the more gratuitous flights of higher theory” as mere“mathematical recreation...without ontological rights”2). She stresses that thedominating mode of development within real life set theoretic endeavor appears to beone of “maximize the range of structures of potential interest,” in sharp contrast toQuine’s crabbed gospel of “minimize and Ockhamize.” All of these observationsstrike me as entirely on target.

She is further troubled by a band of zealots who have set out, armed withpitchforks, to reform the excesses of mathematical postulation under the banner of“naturalism.” Surely, something has gone awry when serious authors fancy that theyassist the Progress of Man by rewriting regularmathematical assertions within weird codes or by hidingsuch assertions behind sentential operators that signify “Idon’t believe it but I want to use it.”3 Later in the essaywe’ll examine the philosophical joy juice that inspiresthese misguided enthusiasms.

Nonetheless, Maddy’s own travels convey her to afinal destination that I find uncomfortable: to an unfetteredrepublic of Pure Mathematics whose governing tenets areentirely determined by the “practices” of its democratically nominated inhabitants--themathematicians. Like most would-be empiricists, I’d rather not allow any portion oflanguage to escape so flagrantly from the tribunal of experience. It is a sad butundeniable fact that all of our “practices,” without adequate responsibility to exteriorcorrective, are apt to fall into the stagnating grasp of cults and/or apriorist philosophy. Even a classificatory predicate of minimal utility such as “contains orgone” can enjoya lengthy and fulsome linguistic life if its employment remains sustained within the“practices” laid down by some mutually reinforcing combination of gurus anddisciples (supplying, perhaps, a psychologically fulfilling “form of life” but not onethat displays an improving empirical grip on nature4). When Wittgenstein complainsof set theory:

In a dark cellar roots grow yards long,5he expresses the concern that even earnest mathematical labors can fall into a trap ofimproperly constrained development. As it happens, I will argue that it is wrong to

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Our computational place in nature

look upon set theoretic practice in this deprecatory manner. Nonetheless, Wittgenstein’s worries capture, in generalterms, the snares of unproductive disengagement againstwhich we should remain ever vigilant, within every field ofendeavor (including mathematics). So I like not Maddy’sappeals to the raw “practices” of the subject.6

What is a “practice” anyhow? Some assembly ofstrategies that we have developed that strive to reach useful answers through someadmixture of measurement and computation. Whether these gambits will trulysucceed or not, at bottom represents a fact about nature that we need to assess, as bestwe can, employing the tools of science. How do we manage to reach theseassessments? How do we manage to work our way out of the clutches of aunprogressive cult? Often, we’ll find, reasonable answers carry us deeply into arenasthat look much like “pure mathematics” of the finest crystal.

So it is here that I propose our own pilgrimage should begin: in weighing whatwe currently know about our computational position within nature. How ably canwe expect to employ linguistic algorithms fruitfully against the background of acomplex natural setting? And through what intellectual means should we expect toestablish those policies reliably?

Plainly part of this appraisal requires a balancedassessment of the inferential tasks that we can’t expectaccomplish within nature’s confines. Like it or not, thereare some outcomes that we’ll never be able to predictthrough calculation, no matter how hard we try. Toparaphrase a noted authority (Mick Jagger) on the limitsof ambition: “you can’t always calculate what you want.” Despite the deceptive assurances of Kantians and like-minded philosophical schools, we possess no a priori

assurance that nature is inclined to trim its behaviors in order to suit the feeblecomputational cloth in which we can hope to dress it.

Nonetheless, learning that we can’t calculate our way out of every naturalexigency needn’t prove particularly depressing. One can design a frictionless pinballmachine with bumpers labeled “0" and “1" where the careening pinball, under mostinitial conditions, will progressively bump out a time series of highly non-recursive 0sand 1s. Through brute calculation, we have no way of replicating this sequence. Sowhat? It seems sufficient that we can understand the physical processes involved,even if we can’t grind out the trajectory with the accuracy required. But what sorts ofknowledge are required for “understanding the physical process involved”? This is a

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relatively deep question, to which we shall return.Three basic themes will animate our journey. (1) Our endeavors to improve

our computational lots in life begin with a limited and rather imperfect set of reasoning skills which we must learn to improve through gradual tinkering fromwithin. (2) “Understanding” a novel physical situation often boils down to assemblinga sort of map (or, more accurately, an atlas of maps) that establish suitable localstrategies as to which outcomes we can expect and which outcomes we cannot expectto calculate whilst in their vicinity. (3) The calculations we can actually carry outusually stand at a far degree of transcendental remove from the target phenomena wehope to track, but we can nonetheless frame more or less satisfactory portraits of howthose relationships of “removal” operate. In my opinion, all of these themes lie latentwithin the thinking of the great “pre-pragmatist” philosopher/scientists7 workingtowards the end of the nineteenth century, such as Helmholtz, Hertz, Mach and themathematicians who worked upon differential equations and geometry such asRiemann and Picard.

(ii)

Let me briskly explain what I mean by (2) and (3) by resurrecting an importantexample from the end of essay 4 (theme (1) will appear in section (iv)): the problemof computing a goose’s flight over the North Pole. But let’s first start with a simplersituation: a goose that flies over a flat plane subject to lateral winds of a specified butvarying strength. And let’s simultaneously bear in mind Lord Kelvin’s hard-headedadmonition to attend to the “numerical reckoning” (= algorithmic computation) thatwe can actually carry out in such circumstances:

In physical science the first essential step in the direction of learning anysubject is to find principles of numerical reckoning and practicable methodsfor measuring some quality connected with it. I often say that when you canmeasure what you are speaking about, and express it in numbers, you knowsomething about it; but when you cannot measure it, when you cannotexpress it in numbers, your knowledge is of a meager and unsatisfactorykind; it may be the beginning of knowledge, but you have scarcely in yourthoughts advanced to the state of Science, whatever the matter may be.8

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Suppose, then, that we possess a pretty good differential equation modeling ofthe factors that affect our bird’s flight path along with appropriate initial conditions.9 The most straightforward means of inferentially extracting information from this weestretch of syntax is to apply a standard numerical method such as the Euler’s ruleroutine sketched in essay 1. What we will grind out on thisbasis is a sequence of roundedoff numbers linked to acharacteristic step size Δt. Pictorically, thesecomputations correspond tofilling in the location squareson a piece of graph paperwhose squares have been set atthe mesh size Δt.Plainly, if we render thesesquares too large, our graphwill not supply a very goodportrait of our goose’s flightafter a small number of Δtcomputational steps. But we can normally presume (I’ll come back to that“normally” qualifier in a moment) that if we choose the Δt mesh size small enough,the computational path we draw on our graph paper will closely resemble our goose’sactual trajectory for a reasonable span of time (we recognize that, eventually, minuteaccumulated errors will spoil our results). We can also normally presume that if wecould obtain the pointwise limit of all of these ever-refining graphs (which I label as“set theoretic” in the diagram for reasons that will become clear later), we shouldasymptotically obtain a limiting curve that will exactly copy our bird’s flight(presuming that we had, indeed, correctly registered all of the pertinent physicalfactors within our modeling equations).

In terms of brute computational capacity, we can rarely calculate the exactconcrete features of such a limiting curve ourselves–the details of exactly how ittwists and turns usually lie as far beyond our ken as the goose’s flight itself (indeed,more so, because at least with our goose we can capture its trajectory on film and witha radar gun). But it is, nonetheless, comforting to know that the limit curve is there(if, indeed, it is–it may not be) because it allows us to form an informative portrait ofhow ably we are capturing our targeted goose behaviors within the concretecomputations we can actually carry out (which, at any given time, will lie at some

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two strategies for potentially improving a computation

upon a dailybasis. Thedeductive roadsto reliable results

within science are not so straightforward, for hard-to-capture nature “hath inclosed my

ways with hewn stone, it hath made my paths crooked.” In my darker moments, Iwonder if the typical logic requirement of our Ph.D programs shouldn’t be eliminatedand replaced by a good course in differential equations.10

In any case, we can often improve the quality of our inferential results byshifting to some alternate computational strategy. Suppose that we have reason tosuppose that our goose tends to fly in approximate circles around its nest, keeping awatchful eye on what transpires there. If we reexpress our governing equation inelliptic coordinates centered on that nest, we may be able to compute more accurateflight paths with less effort than working within our original Cartesian coordinatescheme. Or perhaps our goose’s governing equations closely resemble those for somesimpler avian specimen whose flights follow strictly sinusoidal paths. This perfectnearby behavior may serve as a convenient base that allows us to efficiently computeour own bird’s trajectory as a piggybacking perturbation around the analytic data (=exactly soluable formula) we possess for its cousin bird (cf. the volcano analogy ofessay 5).

When we make conversions of this ilk, we are implementing a strategy forimproving the quality of our inferences (indeed, a simpler strategic decision isrevealed in our mere choice of a mesh size Δt because a poor selection in thisdepartment will result in either rotten predictions or lots of unnecessary labor).

A strategic choice of explanatory pattern is also evident in the manner in which

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we attempt to fill out our graph paper grids. Our gooseproblem is classified (see essay 1) as an evolutionary ODEproblem which usually requires that we employ some formof “marching method” routine to fill out a coordinate gridworking forward in time.11 But if we have instead adopted a“search for an equilbrium” strategy such we applied to aclamped drumhead in essay 7, then we find ourselves fillingout our graph paper in a completely different manner,working from the edges of our domain inward in the “renderself-consistent” mode of a crossword puzzle See essay 1 formore on this).

Insofar as I can determine, the original impetus that has led appliedmathematicians to delicately distinguish between different strategies of descriptiveendeavor e.g., evolutionary versus equilibrium or steady state) is that one jugglesnumbers on graph paper quite differently within the different approaches.

Furthermore, there are a broad range of circumstances (much emphasized in theother essays in this collection) where we are obliged to shift computational strategy inthe midst of a problem whether we want to or not. In essay 4, we briefly consideredthe problems of numerically tracking a goose that flies over a curved Earth. That

underlying curvature forces us replace thecoordinate charts in which we are presentlycalculating our bird’s flight with replacementcharts centered upon new terrestrial localesonce the goose travels too far (we certainlyrequire at least two different maps to cover acircumnavigating goose and we usually getbetter results if we employ more). As wedevelop such a strategy for coverage, we havebegun to fill in what mathematicians call (for

obvious reasons) an atlas of charts for the target manifold (here, the earth). But, aswe’ve already observed, different coordinate charts work better or worse forefficiently computing the flight patterns of different geese, even when they focus uponthe exact same regions, and any adequate World Atlas will contain a number ofdifferent “map projections” (e.g., Goode, Mercator, etc.) that offer differentcomputational advantages to computing areas, great circle paths and so forth. Somathematicians puff up their “atlases” to become “complete” in that they containevery conceivable map projection from which one might hope to select a subsetadequate to cover the earth for ones computational purposes (true: a printed

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“Complete World Atlas” would be quite hefty and is not available in bookstores). When we select a sequence of successive, local coverage maps in this fashion, wehave adopted a further measure of strategy in attempting to resolve a desired question(e.g., “where is this damned bird going?”) computationally, whether that adopting that“strategy” will prove wise or not.

This last remark brings me to the nub of our considerations. We all know, fromthe miserably failed schemes that we have personally assayed, that the mere fact that aparticular assemblage of strategies appears promising is no guarantee that they willpan out as hoped. Whether a given collection of inferential gambits will succeed ornot is determined by how successfully their machinations intertwine with the workingsof the external world, against which our hopes and ambitions count for little. This isman’s enduring computational fate: we may pride ourselves that we know how tocalculate a goose’s path with admirable accuracy while all the time some unforeseendefeater lurks in the natural background (such as the earth’s curvature or the failure ofa Lifschitz condition) that will someday unmask our inferential pretensions asineffective or worse.

Indeed, nature often proves crueler than this, for she may allow us to calculatemerrily on without offering any hint, until long after the moment of computationalcatastrophe has passed, that we have done anything amiss. If we naively program acomputer to track our goose across the earth, it’s unlikely to notice the unhappymoments when it numerically sends our goose through a coordinate singularity (e.g.,the North Pole on a Mercator chart) that thereafter turns all of its subsequentinferential efforts into rubbish (as the old song says, our machines “just keepcomputing on” regardless). Such opacity in our computational procedures raisedserious methodological concerns in the early days of General Relativity when it wasquite unclear what various “funny values from the numerics” indicate. Recognizingthat, in general, curved manifolds need to be covered by piecewise charts doesn’t, initself, instruct us as to where these joins should be placed, but at least it warns us towatch out for such eventualities.12

(iii)

So this is where we will begin our odyssey, amidst the motley of opportunitiesfor effective computation that nature supplies for our enjoyment and exploitation. For it strikes me that a significant part of the natural history of mankind lies in theimperfect policies whereby we steer our way through this inferential morass, pullingourselves up by our bootstraps after a lot of promising gambits have been tried andabandoned. At the end of our journey, I hope that we will recognize that the

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Naturalism preached by Quine and Benacerraf is a false Naturalism to be rejected byevery true-hearted pilgrim of scientism. Accordingly, our plan is to wander throughthe same terrain as Maddy traveled in Second Philosophy (rather as Christina followedafter Christian in the original novel) but with a stronger emphasis on pragmaticconcern. Our hope is to benefit from Maddy’s explorations while reaching a terminusladen less with brute appeals to “practice” and situated somewhat closer to traditionalempiricism of an “atlas of computations” cast.13

Indeed, as I employ it, the phrase “mankind’s computationalplace in nature” represents a simple specialization of the generalnotion of an environmental opportunity in the sense that biologicalorganisms finds themselves confronted with a range of exploitableoptions, according to the physical conditions prevailing at the size

and time scales upon which they live. For example, little inlets along the shoreline ofa stream can act as frequency traps, in that they orchestrate their captive waters intostanding wave patterns in much the manner that a master of the musical jug sets the airwithin his vessel into pleasing congruities. The slow moving coherence of theentrapped water offers the enterprising mayfly an opportunity to perch on the side ofthe cavity and reap delightful delicacies at its leisure.14

The circulating motions within our creek side trap frame a dominant structurein the sense of essay 4 and its coherence provides our insect an opportunity forfeeding not available to bulky objects such as ourselves (even if we wished to eatflotsam). Many marvelous studies on scale in the bioengineering literature have madeit clear that the characteristic “worlds” of humans 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.15

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 thethese biological ideas. Certain frogs might wait until their flying prey execute tightloops before they attempt to catch them. Why might such a policy prove advantageous

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for our amphibian friend, rather than simply attempting to impalethe fly anywhere along its path? For reasons we’ll discuss later,the fly’s outward flight path can often be characterized with manyfewer descriptive parameters than are generally required aroundthe smoother parts its journey. This parametric opportunityallows the frog to frame a better estimate of where the fly is likelyto travel a few seconds hence without excessive demands upon itslimited capacities for observation and reckoning.

In such circumstances, such animals scarcely engage in any significant amountof computation (although our frog undoubtedly executes some analogical surrogate foran algorithm), 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 a well matched mixtureof 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 from school. How can we catch them? Trying to augur their trajectories in most locales involvesdifficult calculations offering unreliable results but they are currently frolicking on ahillside from which they are likely to tumble, so we should calculate where the loci oflowest gravitational potential energy lies and lurk there, waiting to pounce. Here weassume that, over the relevant stretch of the childrens’ history, gravitational attractionwill prove the dominant factor affecting their behaviors, operating in league with

friction to bring them to a temporary equilibrium. But whetherthat strategy will work or not depends on our owncomputational and muscular swiftness and how long thechildren remain stunned. If the little rascals get up too quicklyand rush away, our clever gambit will prove for naught.

I’ve concocted this little parable to illustrate the generalcharacter of the physical circumstances required to underwrite

the success of the various explanatory strategies canvassed in essay 1 (specifically, theadvantages of an equilibrium modeling over a straightforward evolutionary plotting): acomputational opportunity simply represents the physical circumstances that rendercertain flavors of reasoning policy effective.16 Plainly, such circumstances rely uponan accord between natural circumstance and human capacity, which Hermann Lotzeexplained as follows:

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 which

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our avatar

relate 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.17

One of our greatest adaptive advantages over the frog and mayfly traces to the fact thatwe don’t rely so heavily upon the sluggish corrections of natural selection to uncoverand exploit the environmental opportunities that lie about us: we can figure out ourown strategies for catching errant children, thank you. But how do we manage tocobble together these routines in this first place (how do we learn to employ multiplemaps in dealing with migrating geese?) and how do we manage to adjudicate theirreliability later (why do the multiscalar routines of essay 4 supply better results thanpurist bottom up or top down estimations?). Our natural histories are hugely shapedby the answers we supply to these questions, for both good and ill, and, as goodnaturalists-in-the-making, we are obliged to give some account of these shaping factsin very much the same fashion as investigators in so-called “biomechanics” report onthe aptness of a frog’s behaviors with respect to the foraging opportunities available atpertinent scales of size and time.

(iv)

As Lotze remarks, we do not “stand at the center of things” (= we possessneither the perfect observational capacities nor the perfect inferential tools that a well-equipped deity might enjoy), but must cobble our way forward building upon a rathercoarse set of initial instruments. So let us begin where all of our human strategiccapacities begin, within the complex and hard-to-chart routines that our distantforebears developed for the sake of efficient hunting and foraging. In so doing,humans, along with many of our animal cousins, have evolved inferential skills in,e.g., geometric anticipation (when objects A and B collide, where will their contactpoints lie?) whose achievements challenge computational algebraic geometers to thisday. Framing and passing along a foundational skill set of this capacity requires longstretches of developmental time and prolonged and shelteredchildhoods that allow for finely tuned learning. Where humans differfrom most of their cousin animals lies in their remarkable capacities totransfer these reasoning patterns to applications that have not beencontemplated before. Indeed, the swift expansion within evolutionarytime of human reasoning capacity seems explicable only through thisplastic reallocation of fixed algorithmic resources, for everything weknow about the evolution of human intelligence indicates that our brains’ remarkableabilities to cobble together ancestral reasoning schemes for novel purposes lies at the

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center of our abilities to adapt to new environments quicklywithout needing to rely upon the slower processes of geneticadaptation.

Let us remind ourselves of the opportunities we canuncover through mapping an intended area of application into avariant setting. Consider, in this light, the celebrated “imaginedmotion” verification of the Pythagorean theorem. If we attempt to mentally amalgamate the two little squares at thetop through a direct imaginary blending, we fail to see why theresult should coincide with the large square on the hypothenuse. But when we apply the same mental adjustments to the embedded squares in thebottom situation, we immediately recognize that an additive identity must occur (asthe repentant slave trader in the old hymn articulates, “I once was blind but now Isee”). Why are we able to immediately reach a conclusion that eluded us above? Somehow situating the imagined motion within a constant reference square allows ourancestral “theorem anticipators’ to keep better track of volumetric relationships.

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,” where its tangent lines have become mapped to points, we canimmediately see exactly where the curve’s oddities lie–in the sharp singularities thathave now been made manifest. Why? Presumably because our native motionanalyzers keep better track of position change than turning angle when we

circumnavigate a figure visually. Suchexamples explain why mathematicians oftendeclare that a well-chosen transfer of a probleminto a novel allows them to “look at theproblem with a fresh set of eyes.”

Lastly, the caveman example of essay 5comprises a sterling illustration of a fruitful

transfer between a computational routine well adapted to modeling the temporaldevelopment of a process over to circumstances where an equilibrium configuration iswanted (no registration of time appears at all). These formal similarities allowcomputer scientists to take an old “marching method” package off the shelf andreassign its steps to “shooting method” purposes (although we must tinker with theold routines by adding on a further stretch of monitoring for self-consistency). Despite the syntactic surface resemblance of the inferential steps followed within the

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two routines, semantically one deals with information inquite different ways within the two circumstances. Asremarked above, it took a fair amount of time beforemathematicians fully appreciated how different the twocomputational situations were, given the fact that the“shooting methods” began life as an adaptive offshoot ofsimple “marching method” routines.

The long and short of this is that even our most successful stretches ofreasoning will forever carry traces of our aboriginal origins and this impressive butcompromised heritage remains the bedrock of wherever we wander in constructingstrategies for dealing with nature effectively. In an allied vein, Ernst Mach very muchhoped that physics might wean itself from doctrinal dependence upon the “mechanicalideas of substance” that we have inherited from our ancestors, but simultaneouslyconceded that our improving descriptive capacities must always piggyback upon ourhunter-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 the higher biological step, which represents the scientific intellect, restsupon the lower, which ought not to give way under the former.18

But then we need to ask ourselves: with what tools do we uncover the strategicpathways that allow us to transcend our original humble skills and preconceptions anddescribe an external world in less biologically inflected terms? And after we havehoisted ourselves up by our computational bootstraps, how do we ratify, with anyassurance, that we have made wise choices in adopting these altered and transferredprocedures? Answering such questions is central to understanding our actual (and“non-central”) computational positioning within nature.

In many ways, our native intellectual predicament resembles that posed byDescartes in his Sixth Meditation: how do we cobble our way to a just appraisal of theexternal world given that the sensory data with which we must work gets delivered tous dressed in secondary quality garb structured largely for the benefit of swift, on-the-fly decision making (“that hamburger looks a funny purple–I’d better not eat it”),rather than providing a transparent rationale for why that such revulsive conclusionsare warranted (“large colonies of Clostridium botulinum reflect light differently”)? However, in his reconstructive projects Descartes enjoyed the comradery of a

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benevolent Deity who had implanted the basic working principles of nature inDescartes’ head as a priori truths, whereas we can claim no comparable supernaturalassistance. We’ve got to go it all alone inferentially–how should we spin scientificgold from hunter/gatherer straw?

It may illuminate the concerns to follow if we recall (for it is frequentlyoverlooked) that Descartes believed that our mathematical reasoning skills wereinherently limited and incapable of tracking many forms of unfolding physical processthrough their natural developments. Instead, he maintained that mathematicalreasoning per se could match natural process directly only when very specialopportunities presented themselves and that its conclusions should otherwise beregarded as supplying merely analogical intimations with respect to more genericforms of physical development. As we shall later see, much of this deductivepessimism traces to the fact that Descartes did not have the infinitesimal calculusavailabl. e to him and its soon-to-arrive articulation led later writers to a much greaterdegree of descriptive optimism with respect to our abilities to follow nature closely inmathematicized terms. But some of our ruminations in earlier essays suggest thatmatch up between, e.g., differential equations and worldly behavior may not prove assharp as optimistic prophets earlier presumed and we will later weigh our conceptualcapacities for escaping from the descriptive distortions that direct reliance upon basictools of the calculus may engender.

(v)

Before we do, let’s turn to nature’s side of the computational ledger–when willshe reward us for our inferential forays into novelty? Fortunately, she offers anabundance of opportunities where “transferred reasoning” policies supply splendidresults. In particular, let’s consider the general ploy of factoring a complex behaviorinto 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"19

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Fourier factoring

Hamilton-Jacobi factorization

The basic prototype for this behavior is the manner inwhich the integers decompose uniquely into primes, andthe roads to many deep theorems travel through thatrepresentation. So successful was this policy that Gauss,Kummer and their successors artificially introducedmissing “prime factors” into other parts of numbertheory so that imitative inferential pathways could bepursued that quickly led to splendid results. Withinmechanics, Hamilton-Jacobi technique asks whetherthe complex dynamical movements on a base manifoldcan be factored into a set of simple behaviors in whichsingle descriptive variables regularly wind their ways

around their own donuts(= tori, if you prefer). Usually, the originaltrajectories are too entangled to permit such anunraveling, but when a Hamilton-Jacobi factoringproves feasible (or even close to feasible), we learn agreat deal about the target system’s qualitative behavior(e.g., that it is not chaotic).

Finally, this book’s example-in-chief–the Fourierdecomposition of a vibrating string into standing waveeigenfunctions–represents a factorization policy parexcellence.

We should likewise remember factoring strategies can remain important evenwhen proper mathematically established decompositions only occur within territoriesat some remove from the circumstances before us, in the manner of the “perfectlyconical volcanos” of essay 5. Sometimes, indeed, the underlying rationale for afactoring strategy hinges on little more than a division between the descriptive factorswe regard as controllable and those that we can’t. For example, it is common practiceto split a time series (earthquake rumblings, stock price fluctuations, etc.) in adominant (often deterministic) trend plus a stochastic part commonly modeled as“noise.” The first part of this “factoring” selects the relatively stable “dominantbehavior” aspects of the time series that we should bear foremost in mind in planningout a suitable reactive policy, whereas the latter warns us of the probable fluctuations

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around this mean of which we should beware. If our observational and/ormanipulative capacities were finer, we might draw the line between “trend” and“noise” in a different place. But once we decide to split our data into these twocomponents, we will want to apply a rather strange, yet typical, “homogenization”operation to the data we collect. Consider, for example, the popular policy ofmodeling “noise” as Brownian motion (= Weiner process). We start (left side of thediagram) with a sample space of trajectories in which little billiard balls keep changingtheir directions due to random collisions with even smaller balls at a lower scales. Butthis is a messy collection to deal with mathematically, so, moving rightward, weprogressively complicate these broken line trajectories by adding in supplementarysets of collisions with even smaller little balls. And on it goes, until as an infinitarylimit, we reach a fatter sample space consisting in fractal trajectories that change their

directions so often that most them fail to possess well-defined tangents (a substantialmeasure of the set theoretic assurance discussed later in this essay is required to quellour prima facie worries that such loose appeals might never stabilize upon a coherentmathematical object at all). The reasons why we want to walk inferentially throughthe dark corridors of this strange space are exactly the same as our essay 4 reasons forinflating casinos of gamblers to infinite proportions20: only then can stupidmathematics locate the clean descriptive parameters hiding within the stochastic mists(in this case the simple numbers that characterize a “random walk”).

Philosophers often presume that when we split a mountain landscape intodecomposed into perfectly conical volcanos and perturbations thereupon, we havethereby provided a somewhat defective representation of the topographiccircumstances, in contrast to some hypothetical notion of the “true shapes of ourmountains” (I here refer to an example from essay 4). Well, if any “defectiverepresentation” is before us, it surely lies with the “true shape” ascription, not the“conical mountain” part. Surely mountains don’t possess coherent classical “shapes”beyond some low scale of quantum decoherence and so our philosophers’ notion of a“true shape” implicitly extends the classical “shape” notion far beyond its validdescriptive reach. In contrast, the dominating “conical mountain” attributioneffectively captures the vital characteristics around which we should construct our

-17-

foraging strategies for harvesting its vegetative andmineral goodies, with supplementary correctionsbuilt in for the random attitude fluctuations as weexplore the hillsides. Viewed from thisperspective, our seemingly approximativefactoring actually operates as a filter that allows usto discern the main features of the mountainsthemselves cleanly, as opposed to the rubble thathappens to mar their hillsides (the stochasticexamples of the previous paragraph exemplifythese filtering capacities of factorization quite clearly). Indeed, the practical advantages of decomposinglandscapes in this manner are so overwhelming that ournervous systems have already incorporated a largedegree of “noise” filtering into their operations, so thatwe can’t help seeing the “perfect cones’ within our imperfect mountain vistas.

In other words, the fiction that some hypothetical “true shape” report exactlycaptures the grounding properties of our mountain represents a philosophical mistakeakin to supposing (as in essay 7) that Sheriff Donald Duck is solely responsible for allimprovements within the public weal, instead of being abetted by a flock of avianallies operating in collective labor harmony. It animates the frequent reminders bycareful mathematicians that differential equations should not be regarded as the solecarriers of descriptive merit, but merely serve as important way stations along thehighways that lead to useful numerical conclusions. J.N. Reddy articulates thissentiment as follows:

It should be understood that the equations governing a physical problem arethemselves approximate. The approximations are introduced throughseveral sources, including representation of the geometry and boundaryconditions, loads and material behavior. Therefore, when one thinks ofpermissible error in an approximate solution, it is understood to be relativeto exact solutions of the governing equations that inherently contain variousapproximations.21

The common strategy of condensing complex behaviors into singularitiesillustrates such filtering virtues as well. Let’s return to our fly-catching frog for anexemplar. If the insect’s reversing curves are very tight and its trajectories remainapproximately coplanar throughout these maneuvers, the escape paths can be nicelymapped onto two-dimensional curves with cusp singularities as shown. Mathematics tells us that such curves can be approximated by so-called “Puisseaux

-18-

series” whose beginning terms take the form of fractional powers--e.g., terms that looklike ax1/3. In comparison, regular portions of the flight (that is, away from the turningpoint singularity) need to be approximated by the power series expressions familiarfrom elementary calculus courses, of the non-fractional form a + bx + cx2 + ... But itfrequently happens that we can approximate the escape path with a smaller number ofterms in Puisseaux-type situations than for smoother curves where the regular powerseries expansions apply. This representational simplicity offers our frog anexploitable opportunity for swiftly estimating where its victim is likely to travel: waitfor the tight turning points and compute, as quickly as possible, the strength of theundetermined coefficient a within its leading term ax1/3. In contrast, our amphibianfriend would need to study the flight path far more intently to provide the largernumber of estimates required for a power series-based strategy.

Of course, it is highly unlikely that our frog’s brain will actually calculatecoefficients and fractional powers directly, but it might easily frame the requisitedeterminations within the form of a “look up table” stored as adjustable muscularroutine. Nonetheless, our mapping to the cusp curve singularity best explains whyevolution would have fitted out our frog with a fly catching routine of this character (itis common within biomechanics to isolate an optimal strategy first and afterwardsexplicate the departures therefrom in terms of the animal’s genetic and physiologicallimitations). In short, we can’t properly understand adaptive behaviors unless we havediagnosed the strategic environment in which such behaviors emerge. But thecollection of available gambits are mathematically determined by the underlyingphysical facts. In fact, in the cases of, e.g., complex foraging routines, the outstandingbiological mysteries appear to be almost entirely mathematical in their underlyingcharacter.

The distinction between being able to implement a strategy and understandingits working rationale will prove important in section (viii), when we begin toinvestigate the importance of strategy within a reasonable “naturalism.”

In an allied vein, Felix Klein (illustrating some central ideas of Riemann)pointed out that the best route to understanding water flow within a bath tub is to startwith its singularities: that is, the inlets and drains where the water comes in and out. Even if there don’t appear to be any “natural places” where such singularities should

-19-

be situated, we should invent some. For example, the internal behavior of a loaded,linear drumhead can be computed quite effectively through adding up a passel of“influence functions” that tell us how singular point disturbances around the interioraffect the interior (that is, if we clamp only point A with a unit weight, the interior willdistort in manner Γ). Quite often, however, we can obtain better results with fewernodes if we “clamp” ourdrumhead at some distance awayfrom its actual edge, for suchchoices can often capture thedrumhead’s actual boundary-to-interior relationships moreeffectively than a morepedestrian placement. “Out ofcountry” circumstances like thisare rampant throughout appliedmathematics, as the strange policies with respect to “light rays” in essay 8 illustrate.

Once again, these singularity-based descriptions shouldn’t necessarily beregarded as semantically inferior to normal interior descriptions. The shock wavesthat form around airplane wings are commonly modeled within continuum mechanicsas wave front singularities, although the relevant “shock region” formed around thewing tip is of measurable dimensions. Now it is possible (and sometimes useful) to“tame these shock singularities” by adding some viscosity terms to the relevantaerodynamic equations (so that their solutions will remain smooth for all time), butthese adjustments rarely lead to more accurate physical models. Why? Because thereis little empirical basis for a simple viscosity supplement, because the turbulent eventsinside a real shock front involve strong thermal effects that are better acknowledgedwithin the singularity treatment (which, at least, asks us to ponder entropy increase)than within the smoothed over “corrections.”

Such examples merely reiterate our essay 7 tenets that effective informationalrepresentations within science operate in complex “cooperative family” manners thatexploit the background explanatory context in subtle ways. This strategic complexityforces a developing science to continually reevaluate its policies of informationalregistration as it moves ahead: methodologies that formerly appeared simple andstraightforward turn out in retrospect to have rested upon very complex pillars.

(vi)

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One of the central reasons why the thesis that “physics lays out its ‘kind terms’in tidy orbits” is utterly wrongheaded derives from the fact our most reliableinferential strategies for uncovering vital factorization behaviors runs through directlythe territory of infinitesimally centered differential equations (aided, as we shall laterobserve, by significant dollops of set theoretic thinking). This formed the nub of oneof the chief observations of essay 4: to avail ourselves of the great descriptiveopportunities latent in the eigenfunction factorizations of pinned strings and beamsand allied linear systems, we must often follow inferential pathways that lead directlythrough the heartland of infinitesimally focused differential equations, despite the factthat such equations provide a patently faulty account of what transpires on a minutescale within our string or beam (this descriptive overreach, indeed, generates the“greediness of scale” puzzles of the eponymous essay). But this often happens inproductive reasoning: the most effective reasoning pathway connecting points A and Btravels through locations C that lie far aside of the country one originally expected.

We’ll return to the descriptive oddities of standard differential equations in amoment but let us first rehearse the important role their development has played in theevolution of general mathematical thinking and the manner in which they have alsoreshaped our conception of what a “physical trait” is. As we observed in essay 4, theearliest proponents of mathematical physics such as Galileo optimistically hoped thatnatural processes would conform to the limitedset of inferential procedures available withintraditional geometry, but those expectationswere soon dashed by more careful students ofmethodology such as Descartes, who realizedthat physically natural motions will display theperfected shapes studied within traditionalgeometry (Galileo’s “triangles and circles”)only rarely. Indeed, he further believed thatthe much wider array of algebraically circumscribed curves contemplated within hisown “analytic geometry” could not turn the trick either.

Geometry should not include lines (or curves) that are like strings, in thatthey are sometimes straight and sometimes curved, since the ratios betweenstraight and curved lines are not known, and I believe cannot be discoveredby human minds, and therefore no conclusion based upon such ratios can beaccepted as rigorous and exact.22

Why? Because he realized that the intuitive multiplicity of freely drawn curves had togreatly exceed those carved out by algebraic formula.

-21-

As we observed in an earlier essay, he also believed that physics itself wouldoften find itself unable to track an unfolding natural process through all of itsintermediate stages. Among other reasons, he held that all materials are composed ofclosely packed minuscule particles of differing rigid shapes without any spacesbetween them. But what must happen when such a particulate swarm encounters asudden constriction while traveling along a pipe? To maintain a fixed volume, therelevant particles must somehow adjust their shapes through fracture and fusion sothat their collective bulk can smoothly pass through the constriction and speed up. But, clearly, these adjustments must transpire infinitely rapidly and to involvedecompositions and fusions at the level of an infinitesimal dust (no “holes” in theflow were tolerated because vacuums were deemed impossible by Cartesians). Butshould we expect to follow the unfolding details of such infinitely complextransitions with our limited human reasoning skills? No; Descartes opined thatadjustments of this complexity lie beyond our comprehension and labeled them as“indefinite processes.” His disciple Rohault explained:

[Aristotle's followers] did not considerthat equality and inequality areproperties of finite things, which canbe comprehended and compared byhuman understanding, but they cannotbe 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.23

In other words, although such infinitely detailed processes must occur frequently innature and God Himself can readily foresee how they will turn out, we humblehumans, comprising Creatures of a Very Small Brain, must perforce lose theirintricacies within a computational fog that we are incapable of penetrating. SinceGod is no deceiver, we are nonetheless assured that, whatever transpires during thesemurky intervals, all of nature’s inorganic operations will unfold according to the samegeneral policies captured within traditional geometry. But we shouldn’t becomecomputationally greedy; we shouldn’t always expect to inferentially track all of thestages of these complex processes ourselves. Such a guarded perspective represents an early articulation of the theme I shalldub mathematical opportunism: nature offers only restricted occasions wherein wecan fully follow her developing processes with the reasoning tools available to uswithin mathematics. Such cavils, as I stressed in essay 4, needn’t represent any sort of“anti-realism” with respect to the external world itself (Descartes certainly was a

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robust realist), but merely reflect our limited capacities for tracking generic physicalprocesses accurately over long spans of time with mathematical tools. The mere factthat many natural processes evade convenient computational footholds does notautomatically render their workings “inscrutable” in some more mystical manner. Wehave already noted that there are lots of simple processes that we can understand wellenough (earlier I cited a frictionless pinball machine with bumpers labeled “0” and“1”) but where we recognize that rarely will we be able to grind out an accurateenumeration of events as they unfold over time.

If we seek a more optimistic assessment of our descriptive capacities, ourconception of what the subject matter of “mathematics” comprises needs to enlarge. For Galileo, Descartes and their peers, mathematics is regarded as a subject delineatedby the concrete set of a priori tools it provides for reasoning. But these procedures arefully trustworthy only with respect to the rare physical circumstances that suit its strictprerequisites and remain informative about the remainder only in some roughlyanalogical fashion. As long as mathematic’s tools and obligations are regarded in thislimited manner, we are virtually forced into “descriptive opportunism.”24

Descartes’ specific difficulties with fluid flow plainly trace to the fact that hedid not have the notion of differential equation available to him, where such formulasdirectly describe the workings of nature at an infinitesimal level, from which onesubsequently extracts--if one can!--finite scale conclusions through inferentialoperations such as integration or finite difference approximation. The earliestpioneers of the calculus (Leibniz through d’Alembert) were leery of granting theirnewly wrought differential equations “fundamental” mathematical status, because theyworried that such infinitesimally formulas might not make good sense unless theyremained tethered to independently recognized finite figures from which thedifferential equations could be formally “derived” (in truth, some justice lay in theseconcerns–subtle bonds of attachment between big and small are often required). ByEuler’s time, however, it seemed evident that such “finite formula” scruples wereuntenable, for most differential equations required solution sets that lay far outside anyreasonable class of “familiar functions.” Such suspicions were directly engendered bythe fact that the “approximation curves” they could spin out from their equationsthrough techniques such as Euler’s method gave the appearance of closing in on finalresults that looked dissimilar to any curves of prior acquaintance.25 Euler and hisfollowers concluded that suitable differential equations, operating at the infinitesimallevel, could grow their own curves, whether or not the results corresponded to anypreviously encountered mathematical object or not. After all, the differentialequations of Newtonian physics appeared to capture nature’s operations perfectly,albeit only at an infinitesimal level. So shouldn’t mathematics expand its tolerance of

-23-

acceptable functions to suit the gizmos that should represent their mirror imageswithin nature’s realms?

Such are the suggestive grounds of our modern presumption that differentialequations lay out “clouds” of infinitesimal tangent vectors in their own right (i.e,establish a vector field) from which novel finite curves can be obtained by connectingthe little arrows end to end in the manner of a child’s “fill in the dots” puzzle. Withthis acceptance, the question arises: how closely will our concrete calculations (= thesets of finite arrows that we can extract from differential equations through numericaltechniques such as Euler’s method) resemble these infinitesimally generatedstructures? On the basis of bitter numerical experimentation, practitioners quicklyrealized that, with an inadequate choice of step size Δt, one could readily grind outhuge swarms of arrows that provide a completely misleading portrait of what theequation’s limiting curve solutions actually look like (we’ll consider such a case later). Worse yet, ascertaining a suitable choice for Δt is rarely easy and available answersoften insist upon Δt scales completely beyond the range of feasible practicality. Forsuch reasons, numerical schemes like Euler’s were not widely employed in earlyyears, although their “intuitive” techniques still provided the epistemological doorwaythat allowed curves “drawn by differential equations” to enter the hallowed halls of“mathematical objects” as estimable, if hard-to-compute, citizens.

More directly pertinent to the actualcomputational techniques of the early pioneers are theinfinite series expansions they would spin off thesame modeling equations through formal division andallied techniques. But, once again, these admittedlyinformative stretches of syntax are apt to behave inerratic ways as well: refusing to converge at all orrefusing to resemble their target curves until an unfeasible number of terms have beenassembled and so forth. Eventually mathematics had to address these informationalfoibles, separating informative wheat from the misleading chaff as best it could. Aswe’ll see, such obligations led to further enlargements in mathematics’ conception ofits proper dominions.

The salient observation is that nature’s native processes appear to becomputationally removed from our concrete algorithmic capacities in a mildlytranscendental manner--viz., the natural processes stand as the infinitary limits ofnested sets of executable algorithms. Such a viewpoint certainly improves uponDescartes’ gloomy assessment that many physical processes bear no evidentrelationship to our computational capacities whatsoever. If this new picture provescorrect (I’ll call it “the naive differential equation picture”), we’ve greatly improved

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our estimate of what “our computational place in nature” is like. The accompanyinglimit-tolerant attitudes, for the first time in mathematical history, rendered theprospects for a genuine mathematical optimism viable: perhaps every processencountered in nature can be fit to an appropriate set of differential equations (withsuitable side conditions adjoined). Indeed, Euler’s simple infinitesimal equations foran incompressible fluid can resolve Descartes’ pipe problem without much fuss,although calculating the exact details of the turbulent flow near the inset cornersremains a challenging computational task even today.26 But this isn’t surprising; inmost natural applications, the differential equations provided by physics typicallycontain far too many nonlinear terms to yield readily to practical computation. Nonetheless, a “mathematical optimist” holding to the naive differential equationpicture can reasonably claim that nature remains “fully understood” at theinfinitesimal level in all of her workings, even if we lack a robust ability to extractreliable numbers from such equations through available reasoning tools.27

Plainly, attitudes close to this “naive differential equation” picture still animatepopular conceptions of “how mathematical physics operates” within contemporaryphilosophical circles. We’ll return to these issues later, in conjunction with theconsiderations that subsequently unseated differential equations from their formerthrone of descriptive centrality.

(vii)

In essay 5, we criticized the popular philosopher’s mythology of a fixed orbit of“physical kind terms” that arise as logical constructions from the vocabularies inwhich the “fundamental laws” of a subject are couched.28 Such “kind term” conceitsstem from the faulty presumption that, “for epistemological purposes,” we can safelypretend that science can set forth its entire “ontology and ideology” in one grand burstof Quinean “all at once” postulation. Such thinking misconceives real life practiceseverely, for viable physical theorizing must continually locate its importantdescriptive parameters through cross-fertilization drawn from data sources entirelydivorced from its postulates. Thus determining whether a complex behavior can beprofitably factored into simpler sub-behaviors represents a process similar to huntingfor lions or unicorns: critters meeting ones expectations may or may not be available. So how do we uncover these hidden quantities when they, in fact, exist? Initially atleast, we follow the plastic practices of our adaptive ancestors and pilfer and adaptstrategic hints from other walks of life. We then verify that these intimations ofhidden traits bear empirical fruit (I’ll return to my “initially” qualifier later).

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Indeed, profitable forms of factoring have been uncovered through utterlywhimsical forms of “borrowing” from non-contiguous disciplines. The electricalengineer Oliver Heaviside (one of my heroes!) was a great master of this dark art andpilfered excellent algorithms for reasoning with differentiable operators from hisschoolboy familiarity with complex numbers (which he then tweaked to suit his“transferred routine” needs). But more regular roads to reliable factoring travelthrough a core set of modeling equations29 and provide an excellent understanding ofwhy the “physical kind terms” picture is so deeply mistaken. An excellent setting forexamining these “find the factoring behaviors” policies can be found in the Sturm-Liouville procedures which generalize to a much wider variety of systems the basicdecompositional policies that Fourier analysis extracts from a vibrating string in termsof standard of sines and cosines. But the factoring qualities of Sturm and Liouville(technically called “eigenfunctions”) rarely prove the same as applications alter. Thefactoring nodes available within a clamped homogeneous drum head (which obeys atwo-dimensional version of the same equation that applies to our string) look quitedifferent from those of a string and represent the products of Bessel functions ratherthan sine waves. In most circumstances, Sturm-Liouville methods do not land onfamiliar functions at all, but squeeze out novel quantities such as the repositories ofunmusical energy that secretly lurk within the clang of an unevenly manufacturedgarbage can lid. To locate these recondite characteristics, Sturm and Liouville mustsearch for them (like lions or unicorns) employing some proto-set theory techniquesthat we’ll survey later. About fifty years previously, the experimenter E. Chlandi haddemonstrated that striking geometrical patterns could be produced by sprinkling sandon vibrating plates and touching them in one spot or another. After Sturm andLiouville’s results became available, it became clear that by touching his plates in therequisite places, Chlandi had suppressed some of its secret Sturm-Liouville modeswhile allowing the remainder to vibrate on. Guitarists are familiar with themethodology: if we lightly touch a string at its midway point, we drain energy from itsfundamental eigenfunction, allowing only the other overtones to continue ringing. The result is a bell-like tone called a“harmonic.” Just so; Chlandi’s sandsettled in the locales that he hadrendered stationary through suppressingsome of the Sturm-Liouville modesthat usually participate in the plate’sbehavior. Through such tests, later physicists could experimentally verify theunsuspected presence of Sturm-Liouville qualities within a wide variety of targetsystems. Once uncovered, these parameters supply a factoring skeleton key that

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Whac-a-Mole

immediately unlocks the otherwise intractable behaviors characteristic of complexvibratory arrangements. Much of the progress in nineteenth century physics followedthis methodology in one guise or other. 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.30

As it happens, luthiers seek allied insight into the vibrational behaviors ofirregular shapes such as the upper plates of guitars by sprinkling sand on them in thehopes of cognate structural enlightenment. In such applications, however, Sturm andLiouville’s search techniques become stymied due to a guitar’s asymmetric contours(we’ll learn why later in the essay) and are unable to underwrite any firm connectionbetween sand patterns and secret repositories of energy. To the best of my limitedknowledge, it remains uncertain what the stationary sand plots on a guitar face signifyin terms of internal characteristics. It remains entirely possible that factoringopportunities of the desired sort are simply not available within a guitar plate.

Yet factoring the integers into primes is scarcely the only strategic ploy that weshould transfer to physical applications whenever nature accommodates; mathematicsis everywhere full of clever stratagems for teasinginformation out of initially uncooperativecircumstances. Indeed, the task of uncovering animportant set of useful physical properties resembles agame of Whac-a-Mole: you can never be sure wherewithin Greater Mathematicsland the indicators offruitful strategy will first pop up. Observe that the labels that we customarily attach tonewly discovered traits (e.g., “the fundamental mode within the Sturm-Liouvillefactoring of this garbage can lid”) usually derive from the branches of mathematicsthat have suggested their uncovering, rather representing the “kind term” titles directlyconstructed from some “fundamental vocabulary” through grammatical combinatorics.

Locating the requisite Sturm-Liouville traits harnesses a fair amount of what wenow regard as set theoretic thinking in a determinately “transcendental” manner (=lots of contractive squeezing over an ample collection of broken line charts). And thisbrings us to a common philosopher’s conceit that has occasioned substantive

misunderstanding within philosophy of mathematics: thepresumption that the mathematical vocabulary that “physicsneeds” is limited solely to the notions required for an adequateformulation of its axioms or “basic principles” (this contention,often reinforced by Quine, represents the key source of “fixedorbit of kinds” misunderstanding). But those limited resources

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cannot carry us to the chief descriptive characteristics we seek, which commonlyrepresent “dominant behaviors” (vide essay 4) that emerge into prominence only aftera varied collection of environmental conditions lock together to frame an“eigenfunction” factoring or similar descriptive opportunity. But we often can’t framea reliable estimate of their existence (in the sense that we can provide for circularplates but not yet for guitar faces) until we can ascertain how the suggestive domainfrom which we borrowed our descriptive gambit compares structurally to the newapplication. As I’ll stress later, resolving these “structural comparison” questionsoften require large bushels of mathematics harvested from far away fields. As we’vejust observed, many of the syntactic designators employed in physics draw their corevocabulary from the branch of mathematics that suggested their potential utility, ratherthan from terminology cited in their founding axiomatology.31

In short, Quinean presumptions that “mathematics is postulated to suit theneeds of physics” has the true state of affairs nearly inverted, as Penelope Maddy haslong averred. Quite the opposite: mathematics’ investigations of effective strategyacross all of its interlocked dominions continually suggest syntactic constructions thatmight serve physics’s parochial purposes ably, after the usual allotment of adaptivetinkering. But physics generally contains no premonitions beforehand that it “needs”these aids; the terminology generally arrives as a blessing bestowed upon thepractical scientist by the fact that nature has presented us with a computationalopportunity structurally suited to the computational gambit under review. But the firstintimations that such strategic pilfering might work frequently arise within far awayregimens.

We will be able to enlarge upon this “physics looks to mathematics for strategicsuggestion wherever it can find it” theme after we turn to a consideration of how wemanage to understand and evaluate the reliability of a proposed descriptive ploy.

(vii)

But we have plainly arrived at what W.S. Gilbert would have characterized as apretty how-de-do. We accepted the doctrine that certain parts of mathematics mightlie at a “mildly transcendental” remove from our concrete computational capacitieslargely on the grounds that differential equations appear to carve out“transcendentally” “limiting curves” that we can rarely compute directly, but whichseemingly capture exactly the trajectories that nature herself is inclined to follow. Inthis frame of mind was “mathematical optimism” born, severing the tight linkagebetween mathematics and computational reasoning that had characterizedphilosophical opinion earlier. Yet this new “mildly transcendental” acceptance rests

-28-

squarely upon what I dubbed a “naive faith in the descriptive capacities of differentialequations” and, within the preceding essays, we have found subtle reasons forwondering whether this naive picture can be regarded as wholly correct (differentialequations proper don’t seem to capture nature’s behavior as directly as once appearedand require various forms of mollifying correction). On what revised basis can we stillaspire to be “mathematical optimists” with respect to the world about us? Answer: wemust elaborate our picture of the “mildly transcendental” gap that separates ourconcrete computational capacities from nature’s own behaviors.

To appreciate the general manner in which these “elaborations” can beexecuted, let’s return to our goose and the multiple coordinate charts required to plotits flight path. There we began with a familiar object–the earth–and strategicallydetermined how we might be able to calculate effectivelyin its presence (answer: stitch together local coveringmaps through transition maps and correct for metricdistortion by a metrical key within each chart). Mathematicians soon realized that there are a wide arrayof less familiar “manifolds” that can provide “bettersettings” in which all sorts of puzzling mathematicalbehaviors can be transparently unraveled (Riemann, forexample, realized that the strange behaviors of many ofthe “special functions” of mathematical physics become readily understood if they arereassigned to the “setting” of a so-called “Riemann surface”). But how do we assureourselves that the new “manifolds” invoked in this manner qualify as legitimate formsof mathematical structure? (Riemann’s own intuitions sometimes led him astray insuch matters). In one of the most extraordinary developments to emerge fromnineteenth century philosophical thinking, Dedekind, Weyl and their followersdecided to reverse the conceptual dependencies sketched above and introduce thedesired manifolds by characterizing how their intrinsic traits relate, through “mildlytranscendental” ties, to an exterior atlas of covering charts that can be computationallyproduced. If these “transcendental” links of dependency prove well-defined in settheoretic terms, the covering charts will deposit correct traits upon the target manifoldin a wholly trustworthy manner.

Here’s the general idea. Suppose we want a friend to form a correct impressionof our absent Uncle Jeff. If we merely show her an isolated photo or two, she willlikely form a somewhat erroneous picture due to the inevitable distortions that appearwithin any two dimensional photograph. “Is his nose really that big?,” she asks andwe reply, “No, it’s pretty large, I acknowledge, but this replacement photo supplies abetter impression of its magnitude.” But, of course, this new snapshot will be itself

-29-

marred by some further unwanted feature (certain theorems of map projection renderthese distortions unavoidable), so we may need to cover Jeff’s nasal regions with yet afurther chart. And we might also provide our friend with a metrical key that allowsher to compute objective nose length from its image in the photo. Adding photo aboutphoto in this corrective manner, we abstract away the idiosyncratic features we don’twant to attribute to Uncle Jeff himself. Since we’ve notdealt with the back of his head yet, we must supply afurther flurry of charts for its delineation (for the samereasons as the earth requires a connected atlas of maps). But if we eventually surround our uncle with acompleted atlas of every photo that might be possiblytaken of him (complete with transition maps telling ushow to collate the information supplied betweencharts), then we will have removed every occasion forerroneous conclusion on the part of our friend. But a full atlas of the desired stripewill require an infinity of individual photographs.

In just this way, so mathematicians introduce their various flavors of“manifold.”32 They assemble a richwill set of covering charts into an atlas that cancover every sector of the target manifold with some local map or other, along withtransition maps that tell us how to “translate” data from one overlapping chart toanother. Because each covering patch will typically exhibit features (such as anEuclidean metric) that shouldn’t be attributed to the target manifold, we remove thisunwanted structure by puffing up a basic set of covering charts to include every formof equally allowable projection in the “photo album” manner just described.

A standard definition of, e.g., a differential manifold doesn’t completely specifyhow our concrete algorithmic abilities will relate to the events that unfold upon thetarget structure, because they only demand that its defining atlas include curtailedEuclidean spaces through which various curves run. But as we observed with thegoose that flies over a plane, even simple curves within Euclidean spaces may lieconsiderably beyond our concrete computational capacities. If we include Δt graphpaper representations of these skills within an amplified atlas, we generate a fullerportrait of our computational positioning with respect to the target manifold in themanner illustrated. We therefore obtain a full representation of “how the road frommanifold behaviors back to numbers runs” in such applications, completely in thehard-nosed vein that Kelvin recommends.33

“Cooperative family” corrections of the sort contemplated in essays 7 and 8 canbe approached in an allied spirit of abstractive tolerance: numerical assessments ofstress obtained from straightforward computational policies must become convoluted

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with a rash of test function mollifications before the results will adequately reflect thephysical basis for talk of “stress.”

Of course, it’s a long, long way to the differential manifold or the weak solutionstarting from our concrete computational capacities, with ample opportunity formistakenly trusting a finite spat of calculations that haven’t settled into resemblingtheir target behaviors closely at all. In the sections to follow, we’ll review theintellectual tools we employ to gain better control over these unhappy circumstances:how do we ascertain ahead of time whether a given calculation will match its“transcendentally delineated” target behavior ably or not?

[Philosophical digression: Recalling our essay 5 discussion of the blind menand their “understanding” of their elephant, the thinking just surveyed rests upon thephilosophical assumption that we possess an adequate conception of a novel object orproperty once we know how to calculate suitably within its vicinity (roughly, “Bysuch signs ye shall know it”). However, we’ve also observed that such demands needto approached in “mildly transcendental vein: we can possess a general picture ofhow our syntactic computations relate the worldly events they target, without directlyknowing whether a given stretch of practical reasoning is reliable or not. But at leastthis distanced form of computational placement doesn’t seem as discouraging as the“horribly transcendental” relationships posited by Descartes, where nature frequentlyengages in “indefinite” activities over which we possess no computational handleswhatsoever, no matter how distantly removed. Indeed, as long as our portrait of our“computational position within nature” seems clear enough (even if lengthy stretchesof limits and convolutions must interpolate between our concrete capacities andnature’s behaviors), we generally feel that we “understand” the world ably enough(indeed, we feel that we “understand” our pinball machine better after we haveascertained the reasons why we’re unlikely to estimate its numerical outputsaccurately). In the words of a particularly wretched country music lyric, wisdom liesin knowing when to hold ones cards and when to fold them.

In my Wandering Significance, I characterized approaches to “adequatedescription” that emphasize “understanding a target object in terms of ourobservational and inferential capacities within its presence” as pre-pragmatic (=preparatory to, but not fully identical with, the strains that became Americanpragmatism) and attributed such themes in various levels of thoroughness to the greatphilosopher-scientists of the nineteenth century who sought to enlarge the conceptualhorizons within which physics and mathematics operated through philosophicalruminations of roughly this character (sometimes the catch phrase “meaning is use” isemployed in an allied spirit, but our observations about “mild transcendence” warnthat we must be circumspect in how we apply such a rude slogan). But such attitudes

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demand that we resist our natural impulses to think that we “understand conceptsadequately” only if the salient notions strike us as “completely understood” in thesame intuitive vein that leads us to fancy that sighted folks “understand” the conceptredness in a manner completely unavailable to the blind (Wandering Significancelabels such views as “classical”). Indeed, none of our nineteenth century pioneersremained unremittingly stalwart in their pre-pragmatisms and allowed that theunderstanding of nature obtainable through “atlas of calculation” portraiture mightprove somehow inferior to our more robust forms of understanding (such as “red”supplies). Helmholtz: [A] sign need not have any kind of similarity at all with what it is the sign of.

The relation between the two of them is restricted to the fact that like objectsexerting an influence under like circumstances evoke like signs, and thattherefore unlike signs always correspond to unlike influences. To popularopinion, which accepts in good faith that the images which our senses giveus of things are wholly true [of them], this residue of similarityacknowledged by us may seem very trivial. In fact it is not trivial. For withit one can still achieve something of the very greatest importance, namelyforming an image of lawfulness in the processes of the actual world. 34

Through this wimpish “need not have any similarity” concession was the oddphilosophy of “merely structural understanding” born (a compromised doctrine thatretains considerable currency even today). Ernst Mach generally waxed bolder (ifcruder) in articulation: our understanding of an intuitive concept such as “force” or“solid” rests, at bottom, upon our capacities to make observations and reasonappropriately whilst in the presence of a target object.35 The chief differencesbetween these skills and those pertinent to unfamiliar manifolds is merely those ofingrained familiarity and varied technique. Whereas we must deliberately assemblethe skills required for reasoning about manifolds in a self-conscious and single-stranded manner, ancestral heritage and parental training have prepared us to reasoneffectively about solid objects within a terrestrial environment in swift and variedways. From this perspective, the gap between “complete understanding” and “merelystructural understanding” reflects nothing beyond the irrelevant characteristic of “easyfamiliarity.”

My own predilections with respect to “conceptual understanding” are Machianin spirit, but I recognize that overly zealous inclinations of this ilk lend themselves tocoarse articulations, as Mach’s own writings bountifully illustrate. The doleful historyof attempts to rein in descriptive excess through crude and simple tests (“verifiabilitycriteria” etc.) demonstrates that the proper contours of “useful descriptive work” arehard to pin down (except in hindsight, long after the fact). Indeed, we fail to

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appreciate the clouded hazards of the human condition if we don’t acknowledge thesediagnostic difficulties forthrightly. In lieu of any reliable policy for sorting good frombad, even the fanciers of “orgone” must be permitted a long leash, in the off chancethat they may have cottoned onto something useful (history advises us that the nuttiestgambits have sometimes borne the noblest fruit).

In any case, we won’t pursue these complex issues further, except to observethat, from a pre-pragmatic perspective, the complex array of inferential policies onwhich we rely in utilizing “red” effectively within everyday usage lie deeply obscuredwithin the mists of our unconscious mental processing. Teasing out how thesesubterranean skills accord with the dominant behaviors of terrestrial solid objectsrepresents a significant challenge to both brain and materials science, along with asophisticated mathematical appreciation of how inferential tactics we secret employactually work. In this fashion, although “red” remains a word that we readilyunderstand in a natural sense of “understand,” there remains a vital secondary sense inwhich the strategic underpinnings of its usage remains a significant scientific challengethat we don’t adequately “understand” at all.]

(viii)

Returning to the fact that we do not reside “in the center of nature,” we realizethat 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 being ableto ride an bicycle well insures that one appreciates the supportive physics behind theactivity. Once again we find ourselves confronted with blind spots largely attributableto intellectual limitations inherited from our hunter-gatherer ancestors. Consider thefact that we often find parlor tricks involving card guessing puzzling. Of a certainmanipulation of this type, Persi Diaconis and Ron Graham write:

It’s a charming trick and really seems to surprise people. Okay. How doesit work? Let’s start by making that your problem: How does it work? You’llfind it curiously difficult to give a clear explanation. In twenty years ofteaching, we have yet to have anyone give a truly clear story.36

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A contractive strategy

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.37 Thesubject is asked to select a number from 1 to 100 and then asked whether her choicelies on seven proffered cards like the three shown on the left ofthe diagram. From these responses the magician “magically”extracts the correct original choice. On the right we transparentlywitness the underlying “story” of the information processing thatmakes the trick work: the subject has unwittingly supplied directspecifications of the successive digits in her number as registeredwithin decimal notation.

One figures out the murderer in a board game like Cluethrough an allied form of “Twenty Questions” enclosure. A wisestrategy for playing this game should squeeze in on the requiredanswer via a refining set of questions as efficiently as possible, which usually requiresthat the initial questions partition the available search space into blocks of roughly thesame size. Mathematicians say that such strategies operate through contractive (orcoersive) entrapment of the desired answer: we progressively pose inquiries thatultimately squeeze in upon a final answer as a “fixed point” (“Col. Mustard in thedining room with the lead pipe”). Sometimes we must balance our desire to reach acorrect answer as quickly as possible (i.e., employing the fewest number ofquestions) against the assurance that we always reach a right answer even in

unlikely situations and probabilisticstrategies for obtaining answers canoperate more swiftly if we eitheraccept a few wrong answers ortolerate the risk of inconclusivecycling.38

If a well designed scheme ofthis nature is laid out clearly, in themanner of our explanation of Dr.Mike’s card trick, it becomesobvious that the steps in theprocess are contractive: we canreadily see why the progressivequestions asked gradually squeezean already narrowed search spaceinto yet smaller components. Butone can also play Twenty

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Questions or Clue utilizing a dumb methodology that proves redundantrather than coercive–children do this all of the time.

In this fashion, successful magic tricks call upon our intellectualheritage as self-improving aboriginal thinkers in two fashions: (1) thefact that, presented in a suitable disguise, the contractive character of aset of questions may allude us yet; (2) mapped into an alternativerepresentational framework in the manner we applied to Dr. Mike’sroutine, the “clear story” behind the trick can become completely evident. Withrespect 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.39

By the “avenues of thought and logic,” he intends the collection of possible events inwhich the audience attempts to locate the magician’s manipulations, when, in fact,they actually fall within some collection that they have not considered. With respectto the trick that Diaconis and Graham discuss, the “magical” concealment involvesdisguising 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“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 to befooled. 40

To obtain this “beam of light,” the authors must map theconcrete card manipulations within their trick into thepurified settings of graph theory before the appropriate “AhHa! Now I see it” recognition gets prompted. So weconfront a two-way street here: we can often constructmystifying magic routines from easy-to-follow inheritancechains through purposefully hiding their contours under allsorts of psychologically distracting clutter.

Here’s another example of the increased“understanding” that comes with an artfully selected

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transferred mapping. Considered in its own light, the singular point at the center ofthe 2D bow tie curve on the bottom plane of the illustration exhibits puzzlingbehaviors with respect to its points of intersection with other geometrical figures, butthe proper rules become immediately clear if we “blow up” the point into the settingof a 3D curve as illustrated.41 Proceeding in this spirit, the great Italian geometers ofthe late nineteenth century untangled more complicated singularities in a very usefulfashion, but doing so required shifting into spaces of a fairly high dimension. We’llreturn to the 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 “understanding”reflects ingrained limitations that we often surmount by mapping strategically opaquecircumstances into settings for which our hunter-gatherer inheritance has left us betterprepared in terms of strategic appreciation (in the same manner as mapping the tangentsingularities of an algebraic curve into a “point singularity” setting provesintellectually revealing, despite the fact that, mathematically, both phenomena coexiston a symmetrical par). In the history of science, practitioners have frequently learnedto execute an inferential strategy quite capably, long before a transferred mapping toanother setting is located that renders the rationale behind its unfolding strategicallytransparent.

Let’s now examine mathematics’ grand paradigm for“understanding inferential strategy” through mapping to a richersetting: the amazing unraveling of series behavior that Cauchyand his followers achieved through imbedding the real line uponthe plane of the complex numbers. Many of the greatest earlydiscoveries within number theory and differential equations were obtained byextracting infinitely long series expansions through various techniques of formalmanipulation (here “formal” means “obtained through rearranging the formulasthrough ‘subtractions’ and ‘divisions’ in a naive manner”). Unfortunately, theseotherwise stretches of fresh syntax display a disagreeable propensity for supplyingreally rotten answers at unpredictable moments (e.g., they blow up to infinity orapproach correct values very, very slowly). Such “bad spots” make it very hard, indealing with an unfamiliar equation, to know when ones conclusions have remainedon track and when they have shunted into the realms of fancy through plowing througha unforeseen “bad spot” unwisely.

Sometimes the reasons for these miserable behaviors are fairly evident, even toour 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 the partial

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sums that gradually squeeze their results ever closer to alimiting value) but these same considerations break down atthe endpoints. However, we don’t find this failure at -1 and+1 particularly surprising because the original formulabecomes infinite there (and hence shouldn’t possess a propersummation in any case). But why does the analogous series

derived in the same formal manner from the formula 1/(1+ x2) break down at thesesame points, despite the fact that its parent formula is well-behaved everywhere alongthe real line? To understand this strange happenstance, Cauchy recommends that welook further out in Mathematicsland. In particular, we should ponder what transpiresupon the complex plane (a bit later Riemann shifted Cauchy’s loci of strategicclarification to sundry “Riemann surfaces”). Cauchy argued as follows. Given thatconventional power series expansions consist in additions and multiplications, ourseries computations will still make sense over the complex number realm, even ifthough our original differential equations no longer carry obvious physical significancethere. Considered from this point of view, we notice that the partial sums remainconstrained by clear limiting bounds inside the shaded circular region but that two“bad spots” appear at i and -i, in a manner completely analogous to the -1 and +1 blowup points for 1/(1- x2). Accordingly, we should doubt whether our expansion for1/(1+ x2) can be trusted anywhere along the “circle of convergence” that runs through+i and -i. But the real-valued points +1 and -1 lie on this bounding circle, so we havelocated a useful “early warning” signal of problematic series behavior by noting theobvious “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).42

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 plane greatlyhelps us understand the otherwise mysterious computational failures that we willencounter in attempting to compute its values employing standard expansion

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techniques. For example, the topography of a Bessel function’s singularities providesthe skeleton key we require to appreciate the patterns of computational adjustmentthat should be followed in computing the shape of a drumhead correctly via seriesexpansion techniques. Such phenomena are deeply entangled with our earlierobservation that stock philosophical pronouncements to the effect that the singularbehavior of differential equations upon the complex plane “carry no physicalsignificance” are poorly conceived.

Presumably, if we were more godlike in our inferential capacities, we wouldn’trely so heavily upon such “transferred setting” considerations to reveal the strategicsoundness of a given computational gambit. But we are not omnilogical and are easilyfooled by magic tricks and series expansions alike. The only remedy is to transfer ourreasoning policies into a variety 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 sieve structurethat effectively locates a card for us in a magic trick may ably sort out defective partsin an assembly line). So a transferred setting that assists us greatly within physicsneedn’t seem particularly “physical” in its own right.

Besides the raw problems of summation failure, regular series expansionsdisplay an annoying inclination to converge at an extremely slow rate, so that (as weobserved in the “little blip” problem of essay 4) thousands upon thousands of termsmust be added together before the results remotely approach the target behavior. Onecan easily be led into horrendous error (and unwisely programmed computers oftenare) through trusting a partial summation long before its parent series has decided tosettle down to an accurate answer. But reliable estimates on rates of series ofconvergence are often hard to obtain.

Even more annoying from a conceptual point of view (yet truly godsends topractical science) are the so-called divergent series that patently grow infinite in thelong run, yet supply extremely accurate characterizations if one only bothers with thefirst few terms in the expansion. Thus the convergent series that Airy obtained for thescattered light around a rainbow requires a full three thousand terms before its partialsums capture the light patterns correctly whereas the divergent expansion that Stokesdeveloped “gets it right” within its first three terms (and begins to add on rottensupplements thereafter). Why? One gets the distinct impression that Stokes’extraction techniques have somehow massaged the original equation-encoded data in amanner that ably highlights the dominant behaviors (in the sense of essay 4) of therainbow light within its initial terms but oscillates madly thereafter in a vain effort torender justice to the little blips that complicate these primary patterns (rather as the

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governor in an out-of-control steam engine will hunt erratically for a stablizedposition).43

But this impressionistic “explanation” of divergent series behavior relies uponattributing human-like “wishes” to a dumb series expansion! Insofar as I am aware, asatisfactory comprehension of the odd behaviors of these perverse gizmos eludes us tothis day. As essay 4 observes, practical science frequently relies upon their excellent“early term behaviors” as crucial way stations along the inferential pathways that leadto many forms of valuable conclusion–they provide a substantive portion of the“asymptotic stitching” that binds together thefabric of present day science. Yet we don’tfully understand the engines of our reasoningswhen they whisk through these divergentseries junctions. We’ll return to the potentialsignificance of these inferential opacities later.

(ix)

Let us return to the “mildly transcendental” viewpoint we adopted when wepersuaded ourselves that differential equations could inscribe well-behavedmathematical curves that we ourselves were unable to draw or compute, except in theguise of increasingly accurate approximations. We rehearsed the historical evolutionthat led from a Cartesian mathematical opportunism (mathematics can capturephysical behavior accurately only upon special, favorable occasions) to mathematicaloptimism (physical behaviors can always be captured fully accurately throughdifferential equation description, where our parochial computational skills onlyapproximate). This great expansion in mathematics’ horizons was significantlyencouraged by portraits of our “computational place in nature” as illustrated, where acontracting cloud of Euler’s method computations (see essay 1 for an explication)surrounds a cannonball’s trajectory in a “mildly transcendental” manner.

But let’s be careful. When we considered the similarly “coercive” behaviors ofthe reasonings behind Dr Mike’s simple card trick or within a game of Clue, the contraction terminates after a finite number of steps that we can concretely verify. But, typically, an infinite number of improvements are required before our Euler’smethod approximations fully squeeze in on their anticipated “fixed point” targets. Clearly, we must extend our faith in finitely achieved coercions to these widersettings, but can we fully trust our hunches on this score? The pioneers of differentialequations clearly did, but can we persist in hunches of this same intuitive character?

We likewise noted that “understanding” an algebraic curve sometimes induces

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us to “blow up” their singularities into dimensions greater than three and the greatItalian geometers of the nineteenth century reached astonishing conclusions throughsuch techniques. But sometimes the same intuitive “unpackings” led to downrighterror. How can we benefit from the increased “understandings” we achieve throughsuch suggestive “transcendental” transfers withoutreaching unreliable answers at the same time?

Indeed, standard approximation techniques fordifferential equations sometimes misfire as well,often for mysterious reasons that can lead toengineering disaster. Let rejigger the differentialequation that I (tacitly) employed in drawing ourcannonball plots in an innocuous-seeming fashion.Based upon the ball’s initial velocity when it leavesthe cannon’s mouth, we immediately know itscurrent fund of energy E0, as it is presently expressedin an entirely kinetic format in origin (i.e., E0 = ½ mdy/dt|t0).44 Next, consider thefirst order equation that articulates the conservation of energy for our systemthatcorrectly claims that, at every moment, the sum of our ball’s newly acquired potentialenergy (measured by gmy) and its current kinetic energy will remain equal to E0: ½mdy/dt + gmy = E0. But if we apply a standard numerical procedure to thisreplacement formula, we often obtain plots as sketched in heavy black: cannonballsthat mysteriously levitate forever once they have reached their highest crest. No

matter how finely we refine our grid lengthΔt, we continue to get these idiotic results. What has gone wrong?

To answer tis question, we need tostudy more carefully how closely theapproximation strategy we utilizedcorrelates with its target curve. And itturns out that we limit what the “worst caseoutcome” might look like. So let us fix aninitial instant and the step size Δt. We

want to determine how closely our approximation technique will come to thegenerally unknown cannonball trajectory that threads through the leftmost initialinstant. With respect to a “nice” differential equation such as I first employed(md2h/dt2 = -g), we discover that its leading coefficient places significant limitationson where the curve might wander over the Δt interval. This datum (which is called an“apriori estimate” because it depends entirely on the form of the differential equation)

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allows us to draw in little horn-like regions within both target trajectory andapproximation remain trapped over an interval of step length Δt. But if these little“escape horns” open out widely very quickly and we also select an overly lengthy stepΔt, then it won’t require many iterations of our reasoning technique before theconclusions we reach run a significant risk of diverging widely from the target curve. However, as the grid length Δt is continually refined to 0, the little escape horns willeventually enclose our equation’s true solution within a “fixed point” trap. In suchcases, our reasoning will prove trustworthy if we make Δt small enough.

But an unfortunate phenomenon (in the jargon, a “failure of a Lipschitzcondition”) affects our second cannonball plot at its turnaround point: the local escapehorns belonging to our modified energy equation open completely at this point, nomatter how finely one selects Δt (I have sketched these little horns gradually openingin the levitating cannonball chart above). At this juncture our approximation methodloses all of its “coercion.” This pathology allows imposter “solutions” to sneak intoour scenario and confuse our approximation method thoroughly (which, after all, is toostupid to distinguish a “good solution” from a bad one). And this is what went wrongin our second cannonball plotting: a ridiculous “solution” in which our projectile neverchanges its elevation (which obeys our revised “energy” formula as ably as a normalprojectile) has entered the premises and our numerical method loses its contractivehandle on which of these “solutions” it should track.

The moral? That we should follow Kelvin’s advice to set aside unreliableintuition and consider the underlying numerics in a hard-headed way (an aprioriestimate tells us how closely the numbers generated by a computational processcorrelate with the numbers associated with the target trajectory). We haven’t evadedthe necessity of considering these correlational matchups within a “mildlytranscendental” framework, but it is now one where we “understand” a computationalpolicy by studying how numerical relationships behave within very large spaces ofnumerical values. But the natural setting for articulating such “limiting” processcomparisons is set theory working over the real numbers and is often only within itsexpansive realms that we find ourselves able to unfold a computational policy in amanner that allows us to reliably ascertain why a particular inferential strategy is likelyto succeed or fail (here, of course, “reliably ascertain” means “a setting that ourhunter-gatherer minds can adjudicate in a competent manner”). Sometimes suchconcerns unfold quite naturally into very large infinities (as when we attempt tounravel, in Gentzen’s manner, the intuitive reasons why we anticipate that Peanoarithmetic is soundly framed).45

I again stress that, from a “naturalistic” point of view, facts about why certaincomputational strategies work well and others don’t represent as integral parts of our

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“looks like a trap...”

natural history as the facts that steel traps represent a good method for catching bearsand that patrolling the bottoms of steep hillsides sometimes proves an effectivestrategy for nabbing truant children. It is a brute fact of nature that our best guidancefor trusting a novel inferential gambit within subject matters A is to transfer itsunderlying policies into some structurally similar but better understood setting Bwhere we’ve already learned to separate inferential dross from inferential gold in asuccessful fashion. In deference to Frege and the other “logicists,” should we claimthese significant gains in inferential assurance stems from the fact that set theory canbe construed as the “logic of concepts” rendered extensional? I think not; our faith inthe methodology instead rests squarely upon its direct empirical successes inunraveling the enduring mysteries of computational success and failure. ParaphrasingKronecker’s famous remark on the natural numbers, God made the world and Hemade the algorithms, but how they empirically interrelate to one another is man’s task.

In this same vein, I believe that the continuing emphasis, from the days ofFrege and Russell, on set theory’s role in allegedly “providing mathematics with afoundation” has misled many philosophers with respect to its central positioningwithin our scientific investigations. Its chief obligations, it strikes me, are not tosuggest “the important objects that mathematicians should study” (category theoryarguably performs better at that somewhat amorphous task), but to provide avocabulary in which limiting relationships between approximation techniques andtheir target behaviors can be reliably framed and, with luck, adjudicated.

From this point of view, it is important to recall thatthe chief germs of set theoretic thinking entered nineteenthcentury applied mathematics in the guise of the searches forhidden properties that we have already discussed under theheading of “Generalized Fourier” or “Sturm-Liouville”analysis. Although the full details are too elaborate torecount here, the pathways of reasoningthat Charles Sturmand Joseph Liouville pursued in the 1830's locate theFourier-like factoring quantities they seek utilizing severalnetworks of converging approximation of a set theoreticcharacter, prompting the author Hans Sagan to declare of theaccompanying diagram (which locates the salient zeros ofthe successive eigenfunctions):“this looks like a trap and it is.” 46 As I have alreadystressed, the manner in which these authors locate the recondite factoringcharacteristics of garbage can lids bears no relationship whatsoever to standard TheoryT expectations about how science fashions its descriptive vocabularies. The abstract,

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formula-eschewing character of their reasonings is generally viewed as a bellwether ofall modern approaches:

The impact of these papers went well beyond their subject matter to generallinear and nonlinear differential equations and to analysis generally,including functional analysis. Prior to this time the study of differentialequations was largely limited to the search for solutions as analyticexpressions. Sturm and Liouville were among the first to realize thelimitations of this approach and to see the need for finding properties ofsolutions directly from the equation even when no analytic expressions forsolutions are available. 47

Roughly speaking, what they had to accomplish is this. Starting solely from the barestructural facts about an important but yet limited class of differential equationstogether with their manners of boundary condition confinement (originally: fasteneddown everywhere and symmetrically arranged like a rectangular plate or a circulardrumhead), they hope to corral a menagerie of eigenfunction behaviors within a seriesof ever-tightening traps that will conserve individual allotments of energy in themanner of a violin string. But drumheads and organ pipes achieve this goal in a muchdifferent manner than a string. To make progress on this front, Sturm and Liouvilleconfined their attention to problems that factor into one-dimensional circumstancesaddressed individually (this is where the symmetries of the boundaries becomeimportant). Once this assumption has been made, they seek the eigenfunctionbehaviors (corresponding to vibrations of a string in a pure overtone mode) where aninitial configuration of the system does nothing except rescale its own behaviors in aperiodic manner48 (this is where energy conservation enters the picture: the system’sstored energy becomes maximally potential when the system sits at its extremeturnaround point and entirely kinetic when it passes through its zero potential restconfiguration). Sturm located these target states by successively squeezing in on them(according to their number of zero crossings) through a “shooting method” variationon the cannon ball tracking scheme employed above. Liouville then established the“factorization” part of the scheme: he showed that Sturm’s purist are were “complete”in the sense that any arbitrary solution to our target problem can be reconstructed as apotentially infinite sum of these purist states.

In point of historical fact, Liouville remained a bit hazy about what a “limit”should be and this led to various wrong conclusions on his part. Much of thesubsequent genuinely set theoretic approach to the “completeness” of the real line byDedekind and others stemmed from such issues (which is why I characterized Sturmand Liouville’s own work as “ur-set theoretic” in character).

Vital physical quantities such as these are ratified through existence proof of a

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2D versus 3D problems

set theortic character, rather than spun off as combinatorial operations upon syntacticunits.49 In the absence of such a proof, we lack comparableassurance that allied traits can be located within the behaviorsof asymmetrical objects such as guitar faces, even when the“laws” (= differential equations) that govern the interiors ofcircular wooden plates and guitar faces might be exactlyidentical. Why? We noted that Sturm and Liouville wereforced to decompose their problems into one-dimensionalpieces before they could reliably squeeze in on theireigenfunctions. Well, every homeowner plagued by animal invasions knows that it’seasier to capture a squirrel in a living room than a bat for the same topological reasons(I write from bitter experience).

In this non-syntactic manner, many important physical quantities obtain their“natural 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 we have just extracted as “theSturm-Liouville qualities belonging to this system” (although this description needn’tprove unique), but that designation reports upon the search we followed within thestructural hallways of greater mathematics, rather than representing a terminologicalfait accompli of the sort that “kind term” thinking anticipates. By approaching issuesof “definability” in a rather sloppy manner, many contemporary philosophers havefailed to acknowledge the crucial assistance that set theoretic construction plays inamplifying our notion of “physical quantity” to workable proportions. Notice as wellthat any workable Sturm-Liouville decompositions can usefully regarded as maps fromposition space into a dualized “energy space” in which we obtain “new eyes” forrecognizing physical capacities of which we were previously ignorant.

Labeling issues are further complicated by the “describe ones landscape interms of perfect volcanos” phenomena of essay 4: the “overtone analysis” capacitiesthat we attribute to real life violin strings represent rather complex perturbations uponthe perfect string factorizations uncovered through Sturm-Liouville methodology (realendpoints leak, after all). Despite this decline from set theoretic grace, simplelinguistic expediency recommends that we should continue to label the imperfectly“dominating behaviors” of real life strings as “Sturm-Liouville” factors, despite thefact that some erosion in mathematical purity has undoubtedly occurred.

Unlike many of our fellow animals, we can ill afford to wait for the lethargicadjustments of natural selection to select reasoning tools appropriate to the physicalprojects we undertake, so we require descriptive representations that allow us to weedout unworthy alternatives for ourselves (we must be able to recognize that certain

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tools fit our hands, in Lotze’s metaphor). But this chore requires that we possesssome means of representing to ourselves the working strategy underlying theemployment we have in view. And these representations will be “structural” incharacter because we must often evaluate strategic proposals through mapping theminto alternative settings where their working principles bcome clear to limitedintelligences such as ours. In the case of an investigation whether a garbage can lidconceals covert Sturm-Liouville factors or not, our most trustworthy means of backingup the inferential techniques we employ is stretch their sinews upon a set theoreticcanvas and observe the numerical entrapments that follow if we extend our finitecomputational tactics to “mildly transcendental” extremes.

The fact that we gradually improve our lot in life by diligently ponderingstructural analogies in this abstract manner represents one of our key behavioral assetsand must be included in any reasonable characterization of the unfolding naturalhistory of homo sapiens. One of the oddest aspects of the “naturalist” projectsconsidered in our opening section is that they represent endeavors unblemished by anyacknowledgment of the fact that we comprise products of that very “nature.” How hasthis happened? (I’ll suggest a few reasons in my concluding remarks).

At the end of the day, I remain (unlike my fellow pilgrim Penelope Maddy) aQuinean empiricist with respect to mathematics in a generalized “Neurath’s boat”manner (with much “theory T” ballast discarded along the way). When and wherecomputational gambits succeed and fail strikes me as a species of core fact to whichmathematics, along with every other branch of science, remains responsible but whoseultimate disposition remains firmly in the hands of Nature, not ourselves. Noexplanatory policies, however well entrenched, can stand long against her witheringdisapproval, if she so chooses to express herself.

But a continuing aspect of retimbering our boats at sea is that of appraising thestrategic wisdom of the descriptive gambits we piece together and, right now, our mosteffective answers pull us deeply into the structural consideration of “limiting”relationships, for which set theory supplies the natural descriptive vocabulary. Is itcertain that our explications of strategic success and failure must forever rest uponthese exact pillars? No, for recall the “pretty how-di-do” raised earlier. Eighteenthcentury mathematics began to conceptualize its descriptive obligations in an alteredmanner, where the successes and failures of computational illation were investigatedagainst the structural background of a world whose behaviors unfold in a mannerlocated at some degree of “mildly transcendental” remove from the successive stateswe can actually calculate through firm algorithmics. As noted before, the recognitionof this “remove” didn’t make scientists feel that they couldn’t “understand” such aworld (in the gloomy manner of Cartesians unable to penetrate “indefinite”

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processes), because they could still provide general road maps that explicated howlocalized patches of useful approximation could be fitted to nature’s distancedbehaviors. However, some measure of this liberated optimism was undoubtedlyencouraged by the consideration that, for a substantive stretch of time, differentialequations appeared as if they might capture portions of nature’s own workings in adirect and wholly correct manner. As previous essays have indicated, this brightpromise is now somewhat tarnished in the sense that our standard calculus tools oftenneed to get hammered, stretched or smeared before they suit nature’s stern demandson successful inferential anticipation.50 We no longer appear to enjoy the direct accessto behaviors at the level of the point-based continuum in which we formerly trusted(we peer at such events through a glass darkened in mollifications). Perhaps thisawkward retrenchment indicates that our continuing reliance upon set-theoreticunpacking has become slightly out of tune (desafinado, as the old bossa nova says). Surely the core diagnoses of computational success and failure provided withininvestigations like Cauchy’s or Sturm-Liouville’s will remain largely intact underfuture refinements within our investigative toolkit, but that preservation is entirelycompatible with rather wrenching alternations in underlying point of view (vide theremarks on our modern assessments of Newton’s optical “discoveries” in the sectionahead). Consider the enigmatic divergent series expansions mentioned before: theyencode information about physically dominant behaviors in a hyper-efficaciousmanner, yet we lack an adequate understanding of how this coding operates. Perhapstheir residual inferential mysteries are symptomatic of larger misconceptions afoot. Perhaps Lotze was right to worry:

[I]ndeed there is always the possibility that a very large part of our efforts ofthought may only be like a scaffolding, which does not belong to thepermanent form of the building which it helped to raise, but on the contrarymust be taken down again to allow the full view of its result.

If so, I wouldn’t have a clue how to repair the lapses. Finding a variant point of viewthat allows us to keep the vessel of science aright while preserving the nub of thewonderful carpentry supplied by Cauchy, Sturm-Liouville, Picard and their manysailors-in-arms strikes me as a tall order indeed.

But “tall order” or not, such issues can be resolved only through the explanatorytriumphs achieved as we gradually gauge our computational position within the largerframe of nature better. “Practices” have little to do with it, except insofar as theircongealing represents some of the behaviors we need to explain.

(x)

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The story sketched in this essay 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 andto which we remain forever yoked. As noted in our introductory section, somemisbegotten conception of scientistic requirement has unleashed an army of reformerswho maintain that “naturalism”’s honor can be redeemed onlythrough odd programs for rewriting the contents of our scientifictextbooks in peculiar ways. What strange intoxicant has sentthem along this path? Much of it traces to the two scripturesmentioned at the outset: Quine’s From a Logical Point of Viewand Paul Benacerraf’s “Mathematical Truth.”

It is easy to discern where Quine goes astray–it stemsfrom his presumption that it is profitable to think of science as operating in spurts ofall-at-once postulation: at time t, we posit an all-embracing theory T which supplies uswith the fixed vocabularies we should employ while operating within T’s ambit. When T’s empirical fortunes turn against it, we seek a replacement T’ and proceed asbefore. This is a view of science from whose ledgers the ongoing necessities ofstrategic adaptation, innovation and monitoring have been thoroughly scrubbed andquickly leads to the faulty dictum that “the only mathematics that physics requires forits own purposes 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 just observed, thischaracterization has the true interdependencies reversed: physics must continuallylook to general mathematical thinking for novel strategic suggestion and for arenas inwhich clotted reasoning policies can be unraveled adequately enough that their meritscan be reliably adjudicated. Insofar as I can determine, potent advice on either ofthese scores might potentially pop up, Whac-a-mole fashion, within any quarter ofmathematics’ far reaching dominions.

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 own contributionto “naturalistic”ensnarement traces to a more traditional thesis about language thatQuine would have surely 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 that

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Scott

must obtain within the external world for the sentential applications of that vocabularyto 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.51

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] standardview is that logical relations are subject to uniform treatment: they areinvariant with subject matter. Indeed, they help define the concept of“subject matter. The same rules of inference may be used and their useaccounted for by the same theory which provides us with our ordinaryaccount of inference, thus avoiding a double standard.52

Without recognizing that he has done so, Benacceraf in one fell swoop has obliteratedall of the “naturalistic” tasks for mathematics that have concerned us within this essay. All gainful employment removed from view, mathematics assumes the unsettlingaspect of a Zachary Scott-style lounger whose motives and objectives appearsuspicious and “unnaturalistic.” Why? Because our physical claims experience nodifficulties in aligning themselves with exterior truth-values while those ofmathematics cannot tie down their own references in the same direct way. Inconsequence, its sundry “posits” appear weirdly disconnected from the causal bondsthat attach our physical vocabularies to the world.

Indeed, from such a vantage point we should worry: we recognizehow Mildred Pierce earns her money, but Monte Beragon’s sources eludeus.

No pre-pragmatist worth their salt should accept the premise thatour everyday physical vocabulary earns its inferential and referentialcredentials in this direct and easy manner, but I will not litigate that casehere.53 Whatever the situation might be with words of popular discourse, Benacerraf’sclaims surely cannot apply to the physical terminologies of which science is composedand his claims otherwise trade upon the carelessness with which he invokes the term“truth-condition.” As other essays in this collection have observed (I lean particularly

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upon essay 8 here), practitioners in the development of science often find themselvesobliged to scrutinize critically the supportive semantics behind establishedterminologies that have been previously regarded as unproblematic. Thus the latenineteenth century recognition that ordinary light rarely maintains its coherence (phaserelationships) over extended spans forced a reexamination of fundamentals thatinduced writers like Sommerfeld to complain:

Laue’s discovery ... proved the distinction between pulsed and waveradiation to be meaningless, in complete agreement with Rayleigh.54

Sommerfeld is insisting, rather unexpectedly, the the two situations pictured should beregarded as identical at short wavelengths. By “meaningless,” Sommerfeld does notmean “incomprehensible” in any ordinary sense, but simply “can’t be assigned aphysically significant truth value” in the sense of “not supported within the realworld.” He intends to reject the received (and entirely natural) picture of what thephrase “white light is composed of component colors” signifies, under which thedistinction that Sommerfeld dismisses as “meaningless” must make sense, even if thetwo distinguished conditions prove hard to segregate empirically (on a standardpicture, the question “do the amplitudes of the light remain smoothly connected onevery scale of size and never displays abrupt jumps?” is always meaningful). Sommerfeld believes that careful consideration of the lack of empirical support forsuch “truth value” attributions indicates that a significantly revised reading of“composed of component colors” must be adopted (e.g., Norbert Weiner’s “powerspectrum” reading, which, mathematically, represents a rather sophisticatedconstruction).

In such circumstances, I declare that a semantic mimic has been unmasked: asituation where the employment of terminology appears as if its words W attach to theworld in manner M, when, in fact, their truesupportive underpinnings operate in fashionM* and where many sentences sharedbetween M and M* share commonsentential truth-values in application(biologists label such superficial copycatpairings as “homoplastic”).

In contrast to this primary, “foundedin real world attachment” construal of “truth value,” we must also consider “truth-values evaluated according to a specific semantic picture of word/worldattachment.” Indeed, I engaged in such an evaluative task when I claimed that

under the received (and entirely natural) picture of what the phrase “white

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light is composed of component colors” signifies, the distinctions thatSommerfeld labels as “meaningless” must possess truth-values. 55

Indeed, this was exactly the basis on which I demurred from Sommerfeld’s attributionof “meaninglessness”: “unicorn” can’t be utterly without meaning if only for the factthat we can still concoct fairy tales in which suitable exemplars thrive. Appreciatingthe semantic picture that agents entertain in employing a usage56 is vital forunderstanding why they employ words as they do, but such characterizations alonecan’t provide an adequate account of why the parties in question have managed to snagimportant real world truths within their nets of assertion. Consider this run-of-millcharacterization of the inferential pathway that led Newton to his celebrated claimsabout light and color:

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.57

A latter day critic such as Sommerfeld will properly observe that, although most of thediscoveries here attributed to Newton can still be regarded as “true” in the sense of“capturing correct physical data in a linguistically suitable fashion,” Newton’s ownunderstanding of what the phrases “composition,” “dispersion” and “colored light”physically signify has proved to be seriously defective, although he undoubtedly couldhave never uncovered his “truths” had he not labored under these pardonablemisapprehensions. Compare the complexity of these evaluative circumstances withBenacerraf’s naive contention, simplified from that quoted above:

Our knowledge of the world, combined with our understanding of thereferential truth-conditions of p, will often tell us how a given individualcame into possession of evidence sufficient to come to know that p.

But a considerable gap separates a sophisticated modern appraisal of what suitabletruth conditions for “light dispersal” require and those with which Newton himselfoperated. Ignoring that chasm, Benacerraf’s schematic cannot render proper semanticjustice to the complexity of Newton’s accomplishments with respect to “truth.” Newton borrowed his notion of “components” from restricted models (chemical

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Tarski

mixtures, musical overtones) that we now recognize scarcely exhaust the range ofworldly considerations that can ably support a linguistic policy of useful factorization(Weiner’s statistical power spectrum being one of them). So an accurate account of“how Newton came into possession of evidence sufficient to come to know that lightwas composed of colors (in a power spectrum sense)” requires a finer graineddevelopmental narrative than Beneacerraf’s unexampled comtentions suggest. Eventoday, when an instructor introduces beginners into optics, she will frame Newton’saccomplishments in the unnuanced terms of the quoted text, trusting that when shelater explicates the sophisticated underpinnings properly required by “component,”she will not be judged as a liar by her pupils.58 Benacerraf’s presumption thatphysical terminologies obtain their semantic referents in direct ways unavailable toour purely mathematical vocabularies greatly distorts the complex entanglement ofinferential and strategic considerations that inevitably play crucial roles incircumstances such as these.

In my estimation, Tarski and his followers strive to capture the basiccorrelational pictures that currently lie behind the manners in which we use ourwords, including the range of potential counterexamples that we anticipate mightthreaten our reasonings. In particular, logic assigns a significance to “=” in such amanner that the basic Leibniz law inference Fa, a =b :. Fb will prove trustworthyunder any contemplated scenario (this is achieved with a standard “soundness proof”conceived in Tarski’s manner). In an allied vein, the pioneers of the calculus oncefancied that applications of Euler’s method will invariably prove “correct” (= withinacceptable error bounds) if applied to cannonball equations with asufficiently short step size Δt but our levitating cannonball examplewarns that these correlative matchups may go badly astray. Fortunately, by employing set theoretic tools and adding a suitable“Lifschitz condition” prerequisite, a careful study of how a targetequation’s “escape horns” behave (which we can calculate from theformula’s coefficients) allows us to rule out those unhappyeventualities and, if we’re lucky, even establish practicable “worse case scenario”estimates on how widely our computations might potentially diverge from the desiredtarget behaviors at a step size Δt. Applied mathematicians call conclusions of this ilk“correctness results” and I view them as the natural analog to the “soundness proofs”supplied under a Tarski-style regimen.

But the snag lies in the fact that such assessments of inferential reliability onlyhold water insofar as the “scenarios contemplated” within the applied pictureaccurately reflect the physical underpinnings of Newton’s usages. Pace Benaceraff, nosemantic portrait of how physical classifiers relate to their real world correlates arrives

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engraved in stone but merely qualifies as a scientific hypotheses that can be shownerroneous when a close study of the de facto matchups between words and worldreveals that their evolving congruencies have followed an unexpected drummer. Plainly, this occurred with Newton and his “component”; we now see that one cannotvalidly reason about “component colors” in quite the way he did, although he largelyreached “true sentences” with his (pardonably) deficient reasoning tools.

It occasionally happens that, after a semantic recalibration of this character hasoccurred, our normal deductive expectations with respect to humble “logical” wordssuch as “and,” “there is” and “equals” fall by the wayside as well. Here’s a simpleexample that is suggestive of these misalignments. Consider the reasoning pattern:

a = f(c) , a b :. b fcand this “exemplar”:

2 = +4, +2 -2 :. -2 +4.Now take whatever reasoning routine you learned in high school to compute, e.g., +3.5. Your routine will plainly deliver +1.87, not -1.87. But this same methodologywill continue to work for complex inputs and if applied to -1, you will get (2 + -2)/2 rather than -(2 + -2)/2.

But here’s the funny part. If you completely circle theorigin employing the same “rule” for a all along, you findyourself supplying -2 = +4 as an answer when you revisitthe +4 position. What has gone wrong? Well, a context-sensitive feature called an “anholonomy” lurks in thebackground: some of these alleged “identities” can be viewedas correct only locally in the sense that they break down oncethey are applied too widely across the complex plane. Ormore exactly, these calculations should be viewed as sitting on a so-called Riemannsurface and remain valid only so long as one confines one attention to a restricted“sheet” within this surface. Often background contextual effects of this general typesilently pull word applications in unexpected directions and that’s what has happen topoor old “ +4 .”

A physical exemplar of this same phenomenon is supplied by the Foucaultpendulums of our science museums. Set swinging, they slowly adjust theirorientation with respect to the museumfloor and, after a full 24 hour day, failto return to their original position(unless one lives at the equator orNorth Pole). At first acquaintance,

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this phenomenon strikes us as very puzzling. What “force” prevents the pendulumbob from returning to its “original position”? Well, none, actually; you’ve mislocatedwhere the effective forces in these circumstances act. The cohesive attachments thatbind the museum floor to the earth comprise the guilty parties in these circumstances,for they twist the floor to a significantly new orientation over the course of a day,while the pendulum itself continues to oscillate within the same plane with respect tothe fixed stars. But we are so accustomed to adjudicating “same position” in terms ofplacement with respect to floors and other earth-attached frameworks that we don’tanticipate the subtle applicational anholonomy that invades our usage after a sufficientperiod of prolongation.

Does any of this show that “logic” is untrustworthy? Well, it certainly showsthat words that usually capture logical categories sometimes fool us. Perhaps weshouldn’t have treated “ ” as a function sign in logic’s sense, but it sure looks likeone. Perhaps we shouldn’t have understood “=” as logic’s proper identity, rather thansome localized surrogate, but it sure looks like one.

I’ve sometimes found that the following metaphor helps clarify the situation.59 Logic lays out the inferential boulevards of a scientific discipline in tidy, Midwesternorder. Unfortunately, a cruel giant (= real world application) presently looms over thetown and will stomp on anyone who ventures into his shadow. Moral: stay out of theshadow region even if a “soundness proof” guarantees that a “trustworthy” road runsthrough it. As the prophet remarked, logic is eternal, so it can easily bide its time untila future semantic reappraisal allows us to relocate our “ands” and “=”’s within animproved discourse in a manner that allows logic’s traditional prerogatives to regaintheir trust once again.60 And the same limitations in authority pertain to the standard“correctness proofs” for numerical methods, for the correlational portraits upon whichthey are founded often prove irrelevant to the applications in which they are concretelyemployed (unanticipated reappraisals of this stripe have proved a commonplaceoccurrence within the annals of computing).

For this reason, Benacerraf’s breezy assurance that we knowenough about “truth-values” of physical claims to firmly ground“logic” as an impeccable inferential engine proves seriouslymisleading in light of the surprising straying we encounter withinreal life scientific development. One should never claim, without alot of exculpatory provisos, anything comparable to Benacerraf’s“logical relations are subject to uniform treatment: they are invariantwith subject matter.” Such cautions do not render logic “empirical”in any evident sense; they merely indicate that logic’s soundness proofs tell us littleabout reliable reasoning within a target discourse until such time as we have forged

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Pilgrim’s escape fromDoubting Castle

better relationships between its syntax and normal logical operationsAgainst such a potentially revisable background, the standard recursion clauses

of Tarski’s scheme (e.g., “closed sentence ‘φ or ψ’ is true in structure D iff either φ istrue in D or ψ is true in D”) must be viewed as possessing advisory status only: it willbe nice if the “information content” registered within our target grammar accepts aparsing that follows the simple correlational patterns presumed in logic class. Butother practicalities often interfere. First order logic, after all, scarcely supplies theprime inferential engine that propels our practical reasonings forward in real life,which rely more directly upon raw algorithmic computation and the basic geometricalskills we have inherited from our hunter-gatherer forebears. As a result, our localmanipulations of “&” and “=” usually emerge within broader swatches of inferentialroutine and their correlational fortunes remain hostage to the large scale strategies inwhich their activation comes embedded. So it isn’t surprising that we mustsometimes jettison our prior expectations about what the symbol “&” signifies in suchcontexts. We can readily acknowledge that “&” undoubtedly signifies somethingwithin the troublesome usage under review--it just can’t be regular conjunction.

What we can properly say is only this. Tarski hasdeveloped a portrait of how inference rules of a “logical”character can be justified against a posited background ofsimple word/world relationships. As such, we can anticipatethat such conditions (or something approximating to them) willbe commonly met within the real life applications in whichlogical reasoning is extensively employed free of strongcontextual controls. But nothing is assured along this line: thefact that we were taught to employ an “&” as an “and” in ourearly days of language learning provides no firm guarantee that unanticipated semanticturmoil might not disturb the word’s employments later (some subtle contextualanholonomy may twist its usage in unexpected directions). As a result, our “earlyapriori” training (essay 5) bears the same relationships to the descriptive vocabulariesthey engender as parents do to our children. We serve as the founding agencies thatimport words and offspring into this vale of tears yet both must seek their ownfortunes as they negotiate the torturous pathways of nature. In the long run, the syntaxnormally associated with logic should never be approached with the rigid ”uniformity”that Benacerraf expects nor will their de facto correlational relationships ever proveunexceptionally “invariant with respect to subject matter.” Returning to thebroader themes of this essay, we should never wish for a language that irrevocablybinds its physical terms to the world in the rigid manner that Benacerraf anticipates,for such semantic carpentry, were it possible at all, would not construct a linguistic

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1. 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).

2. Pursuit of Truth (Cambridge: Harvard University Press, 1990), p. 95, and “Reply toParsons” in Lewis Hahn and Paul Schlipp, eds, The Philosophy of W. V. Quine (LaSalle: Open Court, 1986), p. 400. I’ve extracted these citations from an excellentdiscussion in Penelope Maddy, Naturalism in Mathematics (Oxford: OxfordUniversity Press, 1997), p. 106

3. 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 some mental comfort from composing anentire novel that assiduously avoided the letter “e.”

4. My sensitivities on this score trace to the tedious experience of living through theAge of Aquarius and enduring hours of earnest drivel conveying no ascertainable

domicile suited to human requirements, which must adapt and diversify in response tonature’s whimsically shifting currents. Instead, we must steer our usages by profitingfrom whatever fresh data we can acquire about the strategies that underpin successfulreasoning across a wide array of settings, in the hopes that one of these might proveadaptable to the descriptive challenges immediately before us.

Accordingly, at the end of our pilgrimage, we can answer Quine and Benacerrafbriskly61: language cannot expect to reach reliable alignment with the physical worldthrough discrete acts of mental intention, scientific postulation or baptismal “causeand effect” connection, but must develop its linkages organically through gradualevolutionary processes held together by considerations belonging squarely to the“Science of Strategy and Structural Transfer.” The limitations of our hunter/gathererheritage permit no developmental alternative. But that “Science,” it would seem, issimply “mathematics” under a different name, for it is hardly evident that thissignificant aspect of “naturalistic study” can be meaningfully segregated from thepursuit of orthodox mathematics as a whole.

Endnotes:

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truth-values whatsoever.

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

6. Maddy assures me that she did not intend her appeal to practices to be so absoluteas suggested here. She merely hopes to staunch the distortions of Quineanindispensability through a direct report on the motivations of the set theorists whowork on foundation. She regards the anti-Quinean approach developed here asentirely complimentary to her own. So be it: my chief objective in this essay is tocorrect Quine’s misapprehensions about set theory’s positive role within science at thehumbler levels of quantity ratification wherein it first proved its muster.

7. They qualify only as “pre-pragmatists” because they strongly emphasize theimportance of achieving pragmatic goals while stopping short of agreeing withWilliam James or Thomas Dewey

8. "Electrical Units of Measurement" in Popular Lectures and Addresses, vol. 1,1883-05-03

9. I’ve tacitly assumed that we are working with an ODE model of a purelyevolutionary character and so no boundary conditions are required.

10. I am reminded of Osborne Reynolds’ perceptive comment in his celebrated , “OnVortex 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 p. 185

Just so: a certain pride in the philosophically ascertainable has lead to a neglect ofconsiderations that might reflect on the infallibility of logic.

11. Properly speaking, we have filled in the “base manifold” projection of such atrajectory in our goose diagram–the temporal path has been projected onto the spatialx/y plane. Observe that proper policies for arranging numbers on a grid become morecomplicated when we consider problems other than the “pure” evolutionary orequilibrium settings considered here (e.g., when constraints enter the picture or themodeling equations change type).

12. Putting this point another way, we can generally supply algorithmic instructionsfor filling out a specific numerical grid in a given manner but we usually lack allied

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criteria for shifting from one such chart to another (such rules can sometimes bedeveloped but our general specification of manifolds does not require them). Accordingly, I claim that the atlas-conception of manifold only supplies us with ageneral picture of our computational circumstances without immediately informingus on where and how we should carry this off. Likewise this picture also suggests ageneral strategy for reasoning about migrating geese (plot their careers upon a set ofinterlocking maps) without providing precise instructions for its implementation.

13. My previous extravaganza (Wandering Significance (Oxford: Oxford UniversityPress, 2006) made some readers (e.g, Christopher Pincock, Philosophica Mathematica18 (2010)) wonder if the “empirical semantics” attitudes expressed there with respectto physical predicates could be coherently extended to mathematical notions as well,given the strongly “normative” role that the latter plays in adjudicating the “posits” ofthe former. One might rather conclude that mathematics serves as the repository ofsome “relativized a priori” in the mode of Michael Friedman and various earlier neo-Kantians. There is no doubt, I think, that Friedman et al. catch the underlying flavorof real life applied mathematics better than most “posit”-addled Quineans (cf. theexchange between Friedman and myself in D. Ross, J. Ladyman and H.Kinkaid, eds,Scientific Metaphysics (Oxford: Oxford University Press, 2013). The present essayaddresses these concerns in a more robustly empiricist manner.

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

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

16. A “computational opportunity” represents a rough analog within reasoning to whatthe psychologist James Gibson calls a “perceptual affordance.”

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

18. Principles of the Theory of Heat (Dordecht: Reidel, 1986), p. 390. Mach wasmotivated to defend the credentials of an abstract “thermomechanics” in the mode ofthe Duhem of essay 2.

19. R.J. Walker, Algebraic Curves p. 204 (verify)

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20. As if the dimensions of Las Vegas weren’t excessive enough.

21. Theory and Analysis of Elastic Plates (Boca Raton: CRC Press, 2006), p. 73.

22. quoted in Mathematics in Civilization p. 263; FIND SOURCE. Such concernsanticipate Jacques Hadamard’s subtler complaint that analytic functions are too stiff tomodel fluid flows that trace to uncoordinated spigots. Cf. my "The UnreasonableUncooperativeness of Mathematics in the Natural Sciences," The Monist 2000, andWS, pp. XXX.

23. Rohault’s System of Natural Philosophy (London: Taylor and Francis, 1988), p.33.

24. Intermediate positions become available if we tolerate infinite series expansions. Under natural assumptions, the Weierstrass approximation theorem tells us that wewill be able to approximate the flow as closely as we’d like over that span with such apower series expression. So far, so good, but here’s the rub (which is slightly subtle). Considered in themselves without any regard to applications, the standard algorithmicrules for “+,” “xy,” generate a natural “flow” within the realms of pure syntax. Thatis, if we start with a single position on a piece of graph paper, our polynomial will tellthat dot how it should “flow” across the paper to the right. If we now continuallyrefine this grid to smaller sizes, the same rule naturally induces a genuine real numberflow upon the limiting grid. But this kind of numerical “flow” will occur no matterwhat the real fluid decides to do. The question we wish to raise is this: do we possessany assurance that this autonomous form of syntactic flow can fully capture thephysical flow we are attempting to model? Jacques Hadamard, to his everlastingcredit in his Lectures on Cauchy’s Problem, pointed out that, in many cases, theanswer is clearly “no”--a numerical flow of this power series character possesses astiff “analytic function” personality that force it to diverge eventually from matchingthe looser fluid flows encountered in nature. For further remarks on these matters seeXXX of Wandering Significance.

25. In the 1840's, Joseph Liouville was able to render these impressions precise. Withrespect to the more general advances discussed here, Chapter 8 of Garrett Birkhoff andUta Merzbach, A Source Book in Classical Analysis (Cambridge: Harvard UniveristyPress, 1973) provides an excellent overview, including the Sturm-Liouvilletechniques to be discussed later.

26. Counterflows.

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27. Poincaré’s celebrated investigations into celestial mechanics only deepened ourappreciation of the potential gulf between feasible calculation and the physical realitywe “comprehend” at the differential equation level. He stressed the fact that quiteoften we can concretely compute the long range trajectories of very simple physicalequations only in the vicinity of certain special opportunities (such as periodic points),leaving us with only a “qualitative” understanding of the chaotic messes that lie inbetween.

28. Such conceptions rest, in part, upon woozy conceptions of “law” because essay 4observed that the standard “standing wave” factorization of a vibrating string’sbehavior (i.e., its spectrum of overtones) applies only through a cooperation with thesalient boundary conditions–for otherwise traveling wave energy wouldn’t getreflected back into the interior of the string. Strings with leaky endpoints don’tpossess the properties required to support a standard Fourier-like decomposition. Yet the boundary condition proviso “the endpoints stay fixed” scarcely qualifies as a“law” in anyone’s book.

29. In fact, modern recastings of Heaviside’s procedures follow a more orthodox“pathway through the modeling equations” routing; it was merely that, before LaurentSchwartz invented his “distributions” (essay 7), some vital parts of the linear equationlandscape required for this journey were missing. See essay 7 for more on this theme.

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

31. “Theory T” thinking typically misunderstands the genuine utilities of axiomaticorganization within science, but I won’t enlarge upon these themes further here.

32. Many specific flavors of “manifold” can be introduced in more direct ways but thestandard 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, alternativeroutes are available, but the Veblen-Whitehead treatment most directly illustrates thepatterns of thinking about mathematical existence that I seek to illustrate.

33. Unnecessary misunderstandings within philosophy of science stem from notbearing these elementary cautions in mind, but I won’t elaborate further here.

34. Hermann Helmholtz, “The Facts in Perception” in Epistemological Writings, R.S.Cohen and Y. Elkanan, Eds. (Dordrecht: D. Reidel, 1977, p. 122.

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35. I must exclude “red” because the historical Mach frequently embracedtraditionalist sentiments of a “only sighted folk can properly understand color” ilk. Presumably, this is because he believed that he could gain a proper conceptual libertyfor physics only through locating its evidential class within a strict phenomenalism. The notion that mental classifiers also need to be reconceived within a pre-pragmatistspirit is largely a theme of the twentieth century.

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

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

38. The search curtailing capacities of probabilistic algorithms are well appreciatedwin this regard.

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

40. Ibid, p. 7.

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

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

43. Robert Batterman, “On the Specialness of Special Functions” BJPS 58, 5 (2006),pp. 263-86. Find better reference.

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

45. Here I an tempted to reframe Maddy’s contention that set theory attempts to“maximize the range of structures of potential mathematical interest” to the possiblyindistinguishable claim that it “seeks to maximize the dominions into which concernsabout strategic rigor can be effectively unfolded.” Set theoretic thinking rarely plays alarge role in suggesting interesting structures to working mathematicians; it merelyserves as a court of last resort in determining whether the relationships to numbersimplicit within such schemes can operate as postulated or not.

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46. 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.

47. 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).

48. 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.

49. 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-Liouville factor)... Introductions of the latter sort qualify as legitimate “definitional extensions” only ifthe implied existence claim can be ratified beforehand. Philosophers who shouldknow better are often careless about “definability” issues within the present context.

50. “Oh, everybody recognizes that physics always idealizes,” complacent folksremark. This is a clear example of a situation where a coarse diagnosis proves worsethan no diagnosis at all. For to properly appreciate what our physical conclusionssignify, we must develop detailed assessments of our “computational place in nature”and that ongoing task demands that we separate out a large number of distinctconsiderations that “idealization” lumps together in a single uninformative wad. If Ipossessed a wand that allowed me to ban a single useless word from the annals ofdescriptive philosophy of science, it might well be this one.

51.“Mathematical Truth” in Putnam and Benacerraf, op cit , p. 413. Why any right-minded “naturalist” would succumb to the temptations of the semantic rigidity thesisis a mystery to me, for the doctrine was precisely concocted by philosophers likeFrege who yearned for a priori weaponry that might remain steadfast through all of ouralgorithmic struggles with nature. But surely a “naturalist” should recognize that ourposition within the celestial frame doesn’t permit such an exalted estimation of ourcapabilities. On the Theory T side of these issues, analytic metaphysicians of thestripe surveyed in Essay 5 will warmly assure us that someday science will supply uswith a perfected Theory T from which all concerns of wobbly reference will beentirely expunged, but I fail to discern the philosophical percentages in betting heavily

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upon a horse that won’t cross the finish line until long after we are all dead.

52. Ibid, p. 411.

53. WS

54. Optics, Otto Laporte and P. A. Moldauer, trans, (New York: Academic, ND), p.353.

55. Note that within these “relative to a viewpoint” contexts, “must have a truth-value” can be substituted freely for “must make sense,” where this same exchange israrely appropriate for our “founded within real world distinctions” understanding of“truth-value.”

56. If they operate under any presumptions at all; sometimes in science it is wiser toremain comparatively agnostic about what in the world our words signify forsubstantive periods of time. Cf. WS, Chapter XXX.

57. R. Kimber, R.W. Greiner and J.C. Heidt, eds. Quality Management Handbook 2nd

edition (Boco Raton: CRC Press, 1997), pp. 290-1.

58. Geoffrey Brooker nicely encapsulates the pedagogical dilemma of an instructor inmodern 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.

59. These examples are extracted from “Of Whales and Pendulums,” Philosophy andPhenomenological Research 82 (1), 2011 and "Semantics Balkanized: Joseph Camp'sConfusion," Philosophy and Phenomenological Research 73 (3) (2007)

60. I’ve borrowed this sardonic take on the old cliché from Oliver Heaviside.

61. It also explains my divergence from Maddy: I employ orthodox mathematical factas a means of explaining why the “practices” of the subject commonly lead to usefulresults.