Theory Facades
Mark Wilson
University of Pittsburgh
Abstract: Many common approximation methods in physics practice "causal
process avoidance" in their operative procedures and such methodologies weave
densely throughout the usual fabric of "classical mechanics." It is observed that
Hume was unable to find any grounding for a robust conception of "cause" largely
because he unwittingly looked in those regions of mechanics where genuine causal
processes had already been tacitly expunged.
I will first describe a form of mathematical schizophrenia that commonly
emerges in the courses of practical calculation and then revisit a familiar
philosophical episode from this point of view. More specifically, I'll argue that
several unexpected forms of mathematical strategizing lie in the background to
David Hume's celebrated worries about causation.
Let me begin with an analogy. In the days of old Hollywood, fantastic sets
were constructed that resembled Babylon in all its ancient glory on screen, but, in
sober reality, consisted of nothing but pasteboard cutouts arranged to appear, from
the camera's chosen angle, like an integral metropolis. Likewise, in scientific work
we often encounter constructions that might be analogously labeled theory facades:
sheets of doctrine that do not truly cohere into unified doctrine in their own rights,
but can merely appear as if they do, if the qualities of their adjoining edges are not
scrutinized scrupulously. That is, a "theory facade" represents a patchwork of
incongruent claims that might very well pass for a unified theory, at least, in the
dark with a light behind it.
From an computational point of view, shifting
betwixt a motley collection of ill-matched pieces
frequently makes good pragmatic sense, for a direct attack
on the full mathematical representation proper to a
physical system is apt to leave the applied mathematician
with naught but a sense of frustrated impotence. Often the
only practical recourse for making inferential headway,
even in very simple circumstances, is to decompose the
system's overall behavior into descriptive fragments
where the intractable complexities of the full problem
become locally reduced to more tractable terms by, e.g.,
throwing away smaller order terms or allied stratagems for reducing mathematical
complexity within short range packets. As this happens, important mathematical
-2characteristics of the original representation may get left behind while alien aspects
creep in. The resulting alterations in mathematical character can easily create a
situation where it is hard to reconstitute the true nature of the original physical
problem simply by examining its covering facade alone: Humpty-Dumpty can't be
easily reconstructed from his shards of shell.
To consider a simple, but characteristic, illustration of this phenomenon,
suppose we have a semi-hemispherical bell welded to a table top. The standard
approach to this problem is to drop the terms that theoretically reflect the thickness
of the bell from its proper governing equations and then, by appealing to the
geometrical symmetries as well, reduce the treatment to a very simple formula that
predicts a constant stress everywhere. However, this approximation cannot be
correct near the table top where the stress in the bell actually rises rapidly.
Accordingly, applied mathematicians enforce a different policy of simplification
near the lip of our dome and obtain so-called "boundary layer" equations that
calculate the stresses near the table quite effectively. In essence, we have covered
our bell descriptively with two patches of different mathematical types: that which
handles the bulk of its interior and the other that treats the narrow ring of high
stress near its rim. Of course, we ensure that the numerical values supplied by
these two forms of calculation match along the circumference where they overlap,
but each of our localized representations leaves out important aspects of the
governing physics that prove of vital importance in the patch next door.
Imagine that we are ignorant of the original equations from which our two
local treatments descend and have learned to describe our bell merely by quilting
together our two patches on a wholly empirical basis. In the history of science,
situations of this logical type commonly arise for a host of reasons, not the least of
which traces to the fact that the mathematics required to formulate the "original
equations" is often quite sophisticated and may lie far beyond the ken of a society's
current practitioners, whereas the derivative covering patches often cobble by with
quite elementary mathematical resources. Human patience being what it is, the
temptation soon arises to pretend as if our facade patchwork provides a wholly
adequate descriptive framework solely on its own terms; that the whole story of the
bell's behavior has been adequately told and nothing further needs to be said. In
this impatient spirit, it becomes common to gloss over the palpable oddities in how
our patches fit together with various forms of rhetoric, including much
philosophical gobblety-gook of a pseudo-positivist ilk, such as "by its very nature,
science traffics in intrinsic idealization" (such themes originally became prominent
in philosophical circles as the side product of late nineteenth century attempts to
-3apologize for lapses in physical argumentation that were later recognized as
attempts to evade more sophisticated mathematical foundations through crude
bridgework1). I once saw a children's book illustration in which the King's horses
and men stood triumphantly arrayed around a patently gimcrack Humpty montage,
daylight streaming through its cracks and with much leftover egg scattered on the
ground. This is the image we should frame whenever we confront the
methodological apologists who attempt to pass off theory facades as unified
theories. In the long run, such evasions of descriptive coherence invariably return
to bite their perpetrators, for generally we must seek the original bell equations
whenever we wish to concoct better fashions of approximation. However,
undergraduate texts in physics often get away with a huge amount of "patchwork
masquerading as an integral theory" hokum, simply because they are not in the
business of pressing calculation to highly accurate levels.
Although it would require a long discussion to make my case, I believe
many current philosophical attitudes, even with respect to topics nominally far
from science, have become misguided by uncritically absorbing too much of this
"passing off a facade for a unified theory" double talk. The net result is a large
body of faulty folklore with respect to science that impedes our ability to think
through basic problems clearly.
Since I can hardly tackle such large themes in a short talk, let me endeavor
to illustrate the sorts of problems I have in mind in miniature, by reflecting upon
background circumstances critical to Hume's celebrated worries about causation.
Here, I claim, we can fairly conclude that our wily Scotsman was unwittingly
victimized by a prominent "facade posing as a unified theory" that has likely
deceived most of us in other contexts as well. The misleading patchwork I have in
mind is the collection of doctrines conventionally labeled "classical mechanics" or
"Newtonian theory." In truth, it scarcely takes a high degree of rigorous scrutiny to
see that many of the descriptive tools assigned by tradition to "Newtonian theory"
do not cohere easily on straightforward mathematical grounds--we shall consider
several simple illustrations later. Nonetheless, the rhetorical engines that stoutly
pretend otherwise babble volubly on, not only in Hume's era but even within the
typical university curriculum of our day. To be sure, a palpable "family
resemblance" is often discernible within scattered Newtonian applications that do
not fit together in strict terms, a potential intimation of formalisms yet to come.
1
Karl Pearson's The Grammar of Science (Bristol: Thoemmes, ND) provides
an excellent case in point.
-4Indeed, given the mathematical complications that the latter typically embody,
wise pedagogical policy often encourages pupils to attend only to the hazy
commonalities and not bother unprofitably and prematurely about rigor and such.
However, the line that divides a sensible policy of "let's let the theoreticians worry
about these mismatches" from dogmatic misrepresentations to the effect that "all of
these treatments apply basic Newtonian ideas in exactly the same way" is a delicate
one and is often transgressed in basic physics education, at the cost of decidedly
fuzzy thinking.
I happen to believe that all "Newtonian doctrine," as it is conventionally
assembled, cannot find a proper "family resemblance" unification while remaining
within the orbit of classical ideas alone, but the whole must instead piggyback, as a
derived facade, upon quantum mechanics. But even if my diagnosis of noninternal closure is too pessimistic, it is undeniable that the mathematical setting
required for genuine classical physics unification needs to be far more
sophisticated than was possibly achievable in Hume's day or, for that matter, at
least two centuries afterward (a fair amount of functional analysis over infinite
dimensional phase spaces is clearly needed). But no one warned poor Hume of
these mathematical foibles; instead, contemporaneous poets like James Thompson
sang Newton's praise thus:
Have ye not listened while he bound the suns
And planets to their spheres! the unequal task
Of humankind till then. Oft had they rolled
O'er erring man the year, and oft disgraced
The pride of schools, before their course was known
Full in its causes and effects to him,
All-piercing sage.2
From such paeans to exaggerated accomplishment, an "emperor's new clothes"
bravado enfolds the topics native to "classical mechanics" in a manner that stifles
doubt even when skeptical criticism is fully warranted.
When I was a boy, there was a program on American television ("What's my
Line?") wherein two imposters sought to persuade a panel of minor celebrities that
they were "Mr. X" instead of the real article, who was also on hand. Within the
2
"To the Memory of Sir Isaac Newton" in Joan Digby and Bob Brier, eds.,
Permutations (New York: Quill, 1985). For a delightful survey: Marjorie Hope
Nicholson, Newton Demands the Muse (Princeton: Princeton University Press,
1966).
-5full spectrum of the Newtonian facade as Hume encountered it, many genuine
examples of causal processes directly treated within the descriptive mathematics
are provided, but these genuine "Mr. X"'s are arrayed without effective distinction
next to approaches that tacitly practice various policies of "causal history
avoidance": mathematical tricks that produce effective answers while suppressing
the very ingredients that express causal operations within a fuller formalism.
Unfortunately, prevailing grandiloquence with respect to Newtonian perfection
strongly encouraged Hume to select as his paradigms of "successful treatments in
physics" two counterfeits that rely, in different ways, upon subtle policies of
"causal history avoidance" and to neglect, as inferior, the exemplars cases where
genuine causal behaviors are directly described by Newtonian procedures. Many
of Hume's characteristic observations on "cause" are greatly affected, I think, by
this unwitting misdirection in favor of non-causal confederates.
As to the true processes, these are nicely illustrated by conventional
Newtonian treatments of planetary gravitational systems and falling cannon balls in
a terrestrial field, processes tidily described by differential equations of so-called
"evolutionary type." But such systems of equations are hard to solve and it is
common to resort to various forms of "causal process avoidance" to simplify the
mathematics. A popular recipe for doing this is provided by a focus on a so-called
"equilibrium" or final rest state, as when we declare to Jack and Jill before their
treacherous descent: "I will make no attempt to trace the convoluted causal process
in which you two will now engage but I can confidently predict the final state in
which you'll land and the condition in which you're apt to find yourselves. So I'll
meet you both at the bottom with a medical kit." That is, Jack and Jill's bruised
outcome can be augured well enough without needing to trace the causal history
that leads them there. In its mathematical guise, such a treatment corresponds to
dropping all temporal terms from a set of equations and then posting a localized
patch over the bottom of the hill. In essence, this trick is formally comparable to
the boundary equations we framed for our bell near its rim. But it is precisely the
terms dropped that embody all of the dynamics pertinent to Jack and Jill's
situation. Such "causal process avoidance" is routine fare in many forms of
mechanical calculation.
As an aside, it is worth remarking that anytime we see an appeal to Poisson's
equation, some allied "avoidance" lurks in the background, a fact that philosophers
-6frequently overlook.3 It would take us too far afield to discuss the situation
adequately, but related problems afflict Bertrand Russell's celebrated critique of
causation in physics4: he wasn't able to locate any because he focused upon
replacement equations from which the notion had already been leached away by
various means. These issues are further complicated by the fact that words like
"cause" and "force" display an uncanny inclination to reinsert themselves into
descriptive contexts from which they have been strategically purged (in a Poisson's
equation situation, we commonly consider the "causal processes" operating upon
small particles by the large bodies "frozen" in their vicinity). In a book just
completed,5 I have argued that this syntactic phenomenon of illicit return is quite
common amongst large classes of descriptive words, but I lack the space to explain
why this is so here. Generally, such reentrant words mean something different
when they reappear in this peculiar fashion, but such shifts are often overlooked at
the cost of the confusions that we witness in, e.g., Russell's discussion of causation.
In any case, descriptive strategies that tacitly appeal to equilibrium can only
describe a limited portion of a system's behavior and, if we incorrectly regard the
ploy as forming part of a seamless account of the system's behavior, we will have
mistaken a portion of patchwork facade for a genuine unified accounting.
As I stated, there are two specific pretenders that divert Hume's attention
from the genuine causal processes displayed in the Newtonian treatment of a
falling cannon ball. Each achieves its misdirection in a somewhat different
manner. Our first faker cites the intuitive notions natural to how we think about
machine design--how we plan effective inventions built of gears, rods and cams--to
discredit the adequacy of the "causation" on display in a robustly causal
gravitational treatment. The Swiss engineer Franz Reuleaux called such notions
"machinal ideas" in his great Kinematics of Machinery 6 of 1875 and I shall often
3
A good example of this can be found in Hartry Field, Science without
Numbers, (Oxford: Blackwell, 1980)
4
"On the Notion of Cause" in Mysticism and Logic (Garden City:
Doubleday, 1957). Cf. Sheldon R. Smith, "Resolving Russell’s Anti-Realism
about Causation," The Monist, April 2000 Volume 83, Number 2.
5
Wandering Significance (Oxford University Press, forthcoming). Chapter
Eight enjoys some overlap with the material presented here.
6
Trans., Alexander Kennedy (New York: Dover, 1963).
-7follow this terminology here. As is well known, Newton's action at a distance
account of gravitation was frequently criticized in his time (and afterwards) as
insufficiently embodying the intelligible "causal processes" we think we witness in
a watch or steam engine fitting (this is the vein of appeal that lies at the heart of the
popular complaint that Newtonian gravitation represents a return to the "occult
qualities" of Mediaeval physics). In truth, Reuleaux's penetrating analysis of the
situation (which seems insufficiently known in philosophical circles) shows that it
is precisely from the realm of standard "machinal ideas" that causation proper has
been effectively purged. In short, our machinal imposter makes a true causal
process look poorly because the former promises wondrous but bogus features that
humble real causation cannot adequately match.
However, before I enlarge upon these observations further, let us inspect the
other Newtonian deceiver on our panel: the standard methodology brought to bear
in Hume's time in the treatment of impact--e.g., the collision of two billiard balls.
Once again, this approach pretends to be cut from the same causal cloth as a falling
cannon ball, but, in hard fact, secretly practices a policy of "avoidance" that
silently skips over the critical causative events. Hume focuses upon billiard ball
impact in his discussion of "cause" and finds no robust notion of causal interaction
revealed there. But this isn't surprising, the standard Newtonian approach
descriptively skirts the crucial events at issue.
Since the methodological rationale for evading genuine causation is easier to
explain in the collision case than in "machinal ideas" circumstances, let me enlarge
upon its parameters first. But before we consider the "cause avoiding" tactics of
traditional Newtonians , let us first survey the daunting array of mathematical
elements that will be required in any direct and rational approach to real life
impact, so that we can more sympathetically appreciate the reasons physicists of
Hume's era needed to treat billiards as they did. (1) Describing the mutually
induced distortions of the colliding balls requires, at a minimum, some means for
treating the induced internal stresses, even though the relevant mathematics (partial
differential equations) had not been invented in Hume's era. (2) Further technical
complications require that some more complicated functional analysis type
framework be employed to extend the partial differential equation picture to
encompass the formation of shock waves and complicated reflections (this is
twentieth century stuff). (3) As a part of (2), some consideration of heat loss needs
to be brought into our picture as well. (4) Boundary conditions need to be
articulated that can be reasonably expected to hold during the interval of contact.
Given the intrinsically abrupt nature of the collisions, these are extremely hard to
-8formulate. Although enormous progress on all of these fronts have been achieved,
it is presently dubious7--it is certainly unclear--whether any mathematical
formalism wholly adequate to our intuitive expectations about what happens in a
billiard collision (even in midair!) has yet been articulated.
In the face of these rather frightful mathematical challenges, how did the
treatment of impact standard in Hume's era unfold? Devised by Newton himself,
the basic trick is to almost--but not completely--cover the history of our colliding
balls with two descriptive patches, one devoted to the balls as they approach the
collision and the other as they scatter away from it. But the actual events of
compression and reexpansion that occur when our two balls contact one another
are set in a little window that our method does not attempt to describe. Instead, it
patches over this hiatus by matching our incoming and outgoing sheets according
to a rule of thumb involving gross energetic qualities and a crudely empirical
"coefficient of restitution." The rough reasonableness of such approximation can
be justified by Jack and Jill-like considerations, but it is plain that we've simply
skirted the chief causal events as we normally conceive them. Indeed, by citing
some important ideas of Euler's involving rigid body behavior, this Newtonian
"blanking out the collision events" picture can improved to handle oblique impacts
with tolerable success and supply predictions more or less adequate to most--but
not all!--standard billiard table events. But, in the long run, the dodge is too crude
to handle the blows encountered in, e.g., sophisticated aircraft design, where the
whole army of causally descriptive mathematics (partial differential equations et
al.) needs to emerge from its prior centuries of facade-style suppression.
This example affords an excellent specimen of a phenomenon to which we
philosophers should be aware. When we hope to ascertain "the world view" of
some branch of physics, we typically look at the mathematics provided in a generic
undergraduate or beginning graduate level text. But commonly the mathematics on
display is merely some patchwork approximate to some framework not provided
(and, possibly, as yet unformulated). Indeed, if we trace the trail of footnotes that
ingenuously declare, "for a more detailed treatment, see...," we often encounter, not
merely more "details," but radically dramatic shifts in mathematical setting
altogether. The upshot is that assessments of "world view" based upon freshman
texts is apt to prove highly misleading.
In point of historical fairness, it is important to understand that most
7
Cf. Werner Goldsmith, Impact (New York: Dover, 2001) and Michel
Frémond, Non-Smooth Thermo-mechanics (Berlin: Springer, 2002).
-9Newtonians did not clearly realize they were evading the crucial interval of causal
interaction because many of them (including Newton himself) also believed that
collisions between balls that retain their shapes throughout a collision needed to be
tolerated as theoretically possible (leading to the old conundrum of what happens
when a perfectly immobile object is struck by an irresistible force, the subject of
both Johnny Mercer's song and a prize essay set by the French Academy in 1724 8).
In particular, it was expected that discontinuous, hard sphere impact might provide
a paradigm for the causative infinitesimals they believed were manifested in a
smooth gravitational interaction (that is, it was common to picture a falling cannon
ball as continually struck by little hammers of gravitational impulse). But all of
our subsequent experience with measures and mechanics has suggested quite
otherwise: the non-smooth interaction can be successfully approached only by
piggy-backing, functional analysis style, upon the smooth interaction, rather than
vice versa. However, we can scarcely expect this to have been appreciated by an
eighteenth century Newtonian (although I believe Leibniz' insistence that "nature
does not make jumps" is quite prescient on these topics). Certainly, these faulty
expectations with respect to infinitesimals supplied impactive phenomena with a
centrality in seventeenth and eighteenth century physics that, by all rights, they
shouldn't have possessed. "All the better to deceive you with, my dear," coos our
billiard ball wolf to his victim Hume.
As we'll see, the faulty dominance given to "machinal ideas" (which mainly
exploit the "rigid body" idea) help make this assumption of perfectly hard
impactive infinitesimals seem irresistible in mechanical tradition.
Thus encouraged by the prevailing, albeit misleading, characterization of
Newtonian physics as, at heart, "billiard ball mechanics," Hume understandably
fastened upon pool table collision as providing the "typical physical interaction."
Accordingly, he found no robust notion of "causal process" revealed there for such
processes had been methodologically excluded from their standard treatment:
Here is a billiard ball lying on the table, and another ball is moving towards
it with great rapidity. They strike; and the ball which was formerly at rest
acquires a motion. This is as perfect an instance of the relation of cause and
effect as any which we know either by sensation or reflection. Let us
8
Cf. Wilson L. Scott, The Conflict between Atomism and Conservation
Theory 1644-1869 (London: MacDonald, 1970), p. 22, and Piero Villaggio, "A
Historical Survey of Impact Theories" in Becchi, Corradi et al.(eds.), Essays on the
History of Mechanics (Basel: Birkhauser, 2003), pp. 223-234.
-10therefore examine it. It is evident that the two balls touched one another
before the motion was communicated, and that there was no interval
between the shock and the motion.9
Note that Hume characterizes impact as "perfect an instance of the relation of cause
and effect as any which we know." He continues:
Were a man such as Adam created in the full vigor of understanding,
without experience, he would never be able to infer motion in the second
ball from the motion and impulse of the first. It is not anything that reason
sees in the cause which makes us infer the effect... .But no inference from
cause to effect amounts to a demonstration. Of which there is this evident
proof. The mind can always conceive any effect to follow from any cause,
and indeed any event to follow upon another: whatever we conceive is
possible, at least in a metaphysical sense; but wherever a demonstration
takes place the contrary is impossible and implies a contradiction.
In particular, the Newtonian has purged all the delicate traits of elastic strain, stress
and heat dissipation required to make rational sense of the concussion and retains
only the incoming and outgoing traits of asymptotic shape, momentum and energy.
These quantities are plainly inadequate to rationalizing the causal processes
occurring within a collision:
But as to the causes of these general causes, we should in vain attempt their
discovery, nor shall we ever be able to satisfy ourselves by any particular
explication of them. These ultimate springs and principles are totally shut
up from human curiosity and inquiry. Elasticity, gravity, cohesion of parts,
communication of motion by parts--these are probably the ultimate causes
and principles which we shall ever discover in nature; and we may esteem
ourselves sufficiently happy if, by accurate inquiry and reasoning, we can
trace up the particular phenomena to, or near to, these general
principles....Thus the observation of human blindness and weakness is the
result of all philosophy, and meets us, at every turn. in spite of our
endeavors to elude or avoid it. 10
Plainly Hume is arguing to a stronger conclusion than simply that the laws of
physics need to be empirically confirmed; he claims that we lack the requisite
9
"An Abstract of a Treatise of Human Nature" in An Enquiry concerning
Human Understanding (Indianapolis: Bobbs-Merrill, 1955), pp. 186-7.
10
Enquiry, p. 45.
-11concepts to make what occurs in the breach intelligible (allied, but milder, themes
appear in Locke, for closely related reasons 11). Hume writes:
[T]he necessary conclusion seems to be that we have no idea of connection
or power at all, and that these words are absolutely without any meaning
when employed either in philosophical reasonings or common life. 12
Indeed, if science could weave no better tale of impact than Newton's coefficient of
restitution story, Hume would be right.
Let us now return to our "machinal ideas" imposter, for, as I've indicated, it
represents the key background factor that serves both to discredit conventional
gravitational interaction as a satisfactory causal process and to encourage the
assumption that abrupt impact can serve as a paradigm for what occurs during a
smooth, infinitesimal interaction. Of course, "machinal ideas" (where the main
objects under consideration are perfectly rigid gears, rods and hinges) have been
prominent within physical thinking since the Greeks and, prima facie, we possess
an admirably intuitive understanding of such devices superior to any we enjoy for
any other sort of physical system. Consider, in this light, a prototypical "four bar
linkage" (the diagram is extracted from Reuleaux). If we turn the crank rod
(marked "a-d") around its base ("a-b"), we
feel that we can perfectly anticipate how
every other portion of the device will
move as well. Superficially, this
circumstance seems like a perfect case of
"determinism," but it isn't, at least in the
conventional "past fixing the future"
sense. What we instead witness is what
might be dubbed "here versus there determinism": a localized movement at the
crank is proscribed along an extended temporal interval, which then supplies a
rational for computing the movements of the linkage away from this locale over
exactly the same time span (formally, we have a singular "boundary condition"
rather than "initial values"). A little thinking shows that the relationship of this
flavor of "determinism" to the conventional sort (a so-called "Cauchy problem")
must be quite subtle and tacitly involve a good deal of suppression of complex
11
Locke approaches Hume's assessment fairly closely: An Essay Concerning
Human Understanding, Book IV, Chapter III (New York: Dover, 1959).
12
Enquiry, p. 85.
-12details through approximation.
When we engage in the design of machinery, a pronounced emphasis on
rigid parts sliding or rolling across each other's surface in a frictionless way
emerges into the foreground for essentially computational reasons, although, on
any realistic mechanical appraisal, such circumstances are hard to implement
physically (parts must be carefully machined, linked and lubricated). Furthermore,
its components can be joined together only in very special ways (specified by the
so-called "Grübler criterion") to allow our contraption to display the internal
mobility exemplified in the "here versus there determinism" of our four-bar linkage
(in other words, our pieces can neither be over- or under-constrained, although
these situations would be the natural norm). To the modern engineer, the phrase
"the theory of mechanism" applies solely to the very special physical systems that
roughly approximate such requirements; from this point of view, there are very few
"mechanisms" proper to be found anywhere in the known universe, except on the
surface of earth (which has become considerably cluttered with such gizmos
through man's handiwork and the activities of natural selection). The special
mathematical feature that makes this limited species of assembly so salient to our
interests lies in the design opportunities offered in their reduced patch description:
by appealing heavily to their rigidity, we are able to move into a very simple
mathematical domain (basically, matrix algebra) that allows us to readily perfect
the performance of any invention we might crudely plot. In fact, it is finding this
improved design that represents the important task for the inventor, not the
accurate predication of its detailed causal behavior. It is Reuleaux's special merit
to have discerned this important fact clearly: that, through a collusion of special
purposes (design improvement) and simple mathematical opportunity (matrix
algebra), the "theory of mechanism" effaces itself from orthodox physics and
becomes a topic of a completely different character altogether. As he observed:
[T]he sense of the reality of this separation [of the theory of mechanism
from general Newton-style mechanics] has been felt not only by engineers or
others actually engaged in machine design, but also by those theoretical
writers who have had any practical knowledge of machinery, in spite of the
increasing tendency in the treatment of mechanical science to thin away
machine-problems into those of pure mechanics. 13
The relationship of "machinal" calculations to the processes actually responsible
for the causal behavior of a mechanism are even more oblique here than in the
13
Kinematics, p. 30
-13billiard ball case.
Reuleaux's explication of why "machinal ideas" loom so large in our
physical thinking, although plainly sound once its contours are brought plainly in
view, were not, of course, appreciated in Hume's era nor, unfortunately, even by
most historians of ideas working today. Instead, the presumption remains abroad
that, in some ill-defined way, appeal to machine-like structure must lie at the core
of classical physical thinking, rather than merely representing an opportunistic
mathematical patch spanning a very specialized subdomain. Or, as Hume framed
the erroneous expectations in his Dialogues concerning Natural Religion,:
Look round the world: Contemplate the whole and every part of it: You will
find it to be nothing but one great machine, subdivided into an infinite
number of lesser machines, which again admit of subdivisions, to a degree
beyond what human senses and faculties can trace and explain. All of these
various machines, and even their most intimate parts, are adjusted to each
other with an accuracy, which ravishes into admiration all men, who have
ever contemplated them.14
But the proper, Reuleaux inspired retort to this "argument from design" is, "Well,
there really aren't many such arrangements to be found in nature apart from those
encountered on earth."
More generally, the notion of rigid body--say, in the form of the
"constraints" so central in Lagrangian approaches--represents an exceptionally
productive, yet ultimately venomous, viper that mechanical tradition has long
clasped fondly to its bosom. This form of reductive trick is quite productive, in that
otherwise intractable problems are commonly rendered viable, but it proves
poisonous in that its alien notions cannot be wholly harmonized with the notions of
internal stress and strain otherwise required in a reasonable approach to
macroscopic bodies. "Rigid body behavior" needs to be regarded as a piece of a
patchwork covering facade, housed within an discordant mathematical framework,
that hovers over the wider range of generic mechanical behaviors merely as a
localized approximation. Worse yet, for the reasons already sketched, once the
rigid body is welcomed as a respectable citizen within the kingdom of mechanics,
it is practically inevitable that the non-distorting collisions of perfectly hard billiard
balls spheroids seem as if they must embody the essence of mechanical interaction,
rather than merely representing pallid and gappy substitutes for the genuine article.
Once we are tricked, as Hume surely was, into assuming that collisions and
14
(Indianapolis: Bobbs-Merrill, 1956), p. 143.
-14"machinal ideas" comprise the core of mechanical doctrine, we will not easily
locate the causal processes robustly embodied within classical physics, because we
will have examined specialized techniques that have been sanitized of true causal
processes from their very inception. 15
15
I began musing about these topics after discussing causation with Sheldon
Smith and reading Graciela dePierris, "Hume's Pyrrhonian Skepticism and the
Belief in Causal Laws," Journal of the History of Philosophy, Vol. XXXIX, No. 3,
July 2001. I am also indebted to the NSF for research support with respect to the
foundations of classical mechanics.