Abstract
Following Jacques Hadamard, applied mathematicians typically investigate their models in the form of well-set problems, which actually consist of a family of applicational circumstances that vary in specific ways with respect to their initial and boundary values (and other forms of “side condition”). The chief motive for investigating models in this wider manner is to avoid the improper behavioral conclusions one might reach from the consideration of a more restricted range of cases. Suitable specifications of the required initial and boundary variability typically appeal to previously established experimental conclusions as to how the target system will behave under a range of eternally applied manipulations of the form “If the conditions pertaining to S were altered in manner M, internal features X would/would not alter” (such claims are called manipulation counterfactuals in the essay and arise in a variety of distinct forms). In his investigations of causal reasoning within other parts of science, our first author (Woodward) has emphasized the conceptual importance of counterfactuals of this nature, for which he was been often criticized by authors of a self-styled “metaphysical” inclination. The purpose of this note is to argue, pace these objections, that closely analogous considerations have long been part of the practice of investigating differential equation models in a sensible way.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
How do we determine whether this claim is true or whether Caesar might have used atomic weapons instead?
- 2.
Sometimes the phrase “well-posed” is employed as an alternative, but it invites an ambiguity that we’d prefer to skirt. In the case of standard initial-boundary value problems, Hadamard lays down three basic conditions: (1) solutions will exist for a certain span of time; (2) they will prove unique; and (3) they will demonstrate behavioral stability under suitable norms, selected according to various further criteria. Often the term “well-posed” focuses upon Hadamard’s third criterion, which we will not discuss further due to its associated technicalities. But similar morals pertaining to counterfactuals will apply here as well.
- 3.
Technical remark: due to the collapsed one-dimensionality of this reduced modeling, the elephant loading itself is usually classified as a forcing condition, rather than a “boundary condition” per se. When we instead model our string as a two or three-dimensional solid, the loading converts to a straightforward boundary condition. For most conventional solids and liquids, their exterior bounding surfaces supply suitable opportunities upon which a worthy policy of what we later call “E versus I effacement” can be reasonably effected.
- 4.
This “free variability” is closely dependent upon a suitable collection of parallel boundary conditions choices. If under most starting conditions, the rope breaks shortly thereafter, Dirichlet-style counterfactuals that begin “if the rope were to stay straight at time t1, then … would happen” will no longer seem germane to the modeling situation before us and needn’t display firmly established truth-values.
- 5.
If we instead know that such energy will leak out of the rope at a particular rate, we encounter a different category of boundary condition called a Neumann condition. The critical feature remains that we can ascertain the leakage rate properties of the endpoints on an a priori basis. Other forms of a priori assurance are also considered in applied mathematics (they are collectively labeled as “side conditions” to the interior differential equations), including interfacial stipulations and the constraints we shall later survey.
- 6.
As remarked earlier, many contemporary metaphysicians maintain that a “grounding in laws of nature” is required to support these assertions, but this rhetoric suggests an intellectual quest that rarely takes place. To be sure, there are probably lots of “laws” that govern the internal behavior of rocks and hooks in ways that can further explicate in great detail why ropes can be more firmly fastened to rocks than jello. But we rarely delve into such ancillary concerns when we worry about elephants on tightropes. Merely knowing from experiment that the rope will hold on both ends usually suffices for our purposes.
- 7.
In the jargon of the mathematicians, the side conditions appropriate to a well-set elliptic problem (the soap film) differ from those appropriate to hyperbolic circumstances (the elephant on a rope). For more details on these distinctions, see any standard text on partial differential equations and/or [3]. It should be remarked that engineers commonly approach mild elephant-on- a-rope problems in an equilibrium-centered manner in which they only attempt to ascertain the final shape of the rope after the elephant has settled into quiescence, not attempting to ascertain how it will bounce around beforehand. In such circumstances, the associated well-set problem becomes elliptic.
- 8.
Consult the essay on Pierre Duhem in [3] for more on these experimental advantages.
- 9.
His chief illustration [1952] is quite substantive, for he shows how a parallel limitation to analytic initial data fails to reveal the underlying processes within a hyperbolic modeling. In our toy substitute, the restriction on initial conditions turns off the leftward heading component within d’Alembert’s general solution for the wave equation A(x − at) + B(x + at).
- 10.
For allied reasons, we might want to examine our rope + elephant model over a wider arena, in which we vary the weight of the elephant as a “control variable.” As remarked in an earlier note, forcing conditions sometimes become true boundary conditions when the dimensionality of an example is enlarged. For such reasons, control variable problems concerning boundary condition assignments are often important. Nonetheless, we must distinguish these motives for considering wider assignments of boundary condition values from the more central requirements that reflect the effacement-from-environment concerns we shall detail in the next section. Mathematicians distinguish “control problems” from regular “well-set problems” for exactly this reason.
- 11.
Mathematicians call these stipulations “distributions” (or something fancier, if modeling requirements require). In such circumstances, neither P(x, t0) nor V(x, t0) can be credited with normal numerical values.
- 12.
See “Semantic Mimicry” in [3]. For novelty’s sake, the diagram illustrates a string composed of two sections (gray and black) welded together, causing a finite change in wave speed when the join is transversed. “Side conditions” pertinent to interfacial transport naturally emerge in such contexts.
- 13.
To be sure, certain forms of Neumann condition call upon principles such as “Newton’s law of cooling,” although such provisos rarely satisfy the “law of nature” expectations of the metaphysicians.
- 14.
This energetic redistribution is governed by further physical factors that are not captured within the wave equation proper.
- 15.
Indeed, a more detailed study of violin tone requires that the waves passing through the bridge and nut to the instrument’s body should be scrutinized more closely, at which point our former Dirichlet endpoint condition will open out into a very complex process involving wave equations rather like the one we applied previously only to the string.
- 16.
Although we have selected Hiddleston as our chief target due to the pithy manner in which he articulates his claims, the presumptions he articulates—viz., that counterfactuals require modal “backing” stemming entirely from “laws of nature”—have been widely accepted for decades within philosophy. From this vantage point, philosophical accounts that appeal to “undischarged” (= unanalyzed) counterfactuals are dismissed as inadequately “grounded.” These popular prejudices follow from (or, at least, are naturally suggested by) “metalinguistic” accounts of counterfactuals in which a counterfactual qualifies as true only if its consequent is derivable from its antecedent in conjunction with other premises, including the applicable “laws of nature.” Accounts of this sort trace back to Goodman [1], but broadly similar assumptions have been defended more recently by Maudlin [8] and by Paul and Hall [9], where the “grounding laws” are now assumed to adopt a more specific form—they must represent laws of temporal evolution, and the systems to which they apply should constitute well-posed initial value problems. These apparently represent the background doctrines to which Hiddleston tacitly appeals. The major alternatives to these metalinguistic accounts invoke “similarity relations” among possible worlds in the manner of Lewis [10]. These alternative treatments also assign a preeminent role to “laws of nature,” without special regard for boundary conditions or the other ingredients of normal scientific specification. Many authors within these schools further believe that the grounding “laws” themselves can be reduced to Humean claims about “actual” regularities, leading to the conclusion that all counterfactuals can be assigned fully “actualist” truth-conditions. We firmly contend that none of these purported reductions have been adequately established.
It is worth noting that a number of divergent motivations have been offered for these basic “grounding” assumptions. Some writers (e.g. [11, 12]) articulate epistemic concerns—they maintain that counterfactuals cannot be reliably assessed for truth or falsity without information about grounding laws. For example, the second article mentioned criticizes Woodward’s use of interventionist counterfactuals on the grounds that we cannot determine which interventions are possible and which results would follow if they were to be carried out unless we already know the laws governing the system in question. But this claim is surely false—we can discover which interventions are possible and what happens when they are performed simply by doing experiments and performing manipulations. As we stress elsewhere in the paper, the great effectiveness of Lagrangian methodology within engineering traces precisely to the fact that we can learn the constraints that restrict a system’s movements through direct manipulation without gaining any further information about any pertinent underlying laws.
Other writers (e.g. [8], but also [11]) appeal to broadly semantic concerns—they claim that stand-alone counterfactuals without grounding laws are commonly vague, context-dependent and unclear in a manner that makes them unsuitable for use in science. Providing backing laws is required to repair these deficiencies in truth-condition. We agree that some counterfactuals exhibit these flaws, but these criticisms rarely apply to the manipulationist counterfactuals under review in this essay.
In more recent literature, it is often acknowledged that such epistemic and semantic contentions are unconvincing, and current fashion directly appeals to considerations of an overtly “metaphysical” nature. Often these replacement doctrines are articulated in manners that we find unedifying: viz., “counterfactuals cannot be barely true, but require grounding in what is actual.” Insofar as we can determine, these misty claims stem from the same methodological prejudices as motivated the epistemic and semantic concerns of former times. It is striking how “grounding” doctrines continue to thrive even as their philosophical underpinnings shift significantly in the interim.
- 17.
Contemporary metaphysical opinion frequently presumes that all of the vital traits pertinent to a target system S can be grammatically constructed from the “fundamental qualities” allegedly appearing in the “laws” that govern S (or within their justificatory underpinnings). As [3] points out in considerable detail, this assumption is naively framed. The standing wave tonal spectrum of a violin string plainly represents one of its most important physical characteristics (natural selection, after all, has fashioned our ears and brain to filter away the extraneous noise that surrounds these vibratory characteristics within everyday life). But these traits do not appear as grammatically constructible vocabulary within the relevant “system law” ∂y2/∂t2 = c∂y2/∂x2; indeed, that formula doesn’t pretend to capture all of the physical factors responsible for making standing wave behaviors prominent in our musical lives. As noted above, those traits only become manifest when our interior wave equation is hooked up to boundary conditions that assist the internal energy storage characteristic of tonal vibratory behavior. In terms of present distinctions, the tonal characteristics qualify as a variety of “generalized coordinates.”
- 18.
Note that these manipulative experiments again yield counterfactuals that are not grounded in laws in the sense at issue in this essay: we don’t need, to appeal to laws to explain what the counterfactuals mean or how they can be reliably known and there is no reason to think there is a conceptual link of some kind between the counterfactuals and grounding laws.
- 19.
Many authors cheerfully cite “all ravens are black” and “all dry matches ignite when struck” as candidate “laws,” despite the fact that neither of these assertions look like plausible “laws of nature” in any traditionalist sense. Historically, the notion of “law” emerged within the annals of science in a wide variety of highly irregular ways, often carrying along fossilized remnants of archaic conceptions of scientific method. Significant confusions can arise from the common practice of presuming that some scientific claim enjoys certain formal features simply because somebody long ago decided to label it as a “law.”
- 20.
We write “presumptively,” because simple “n equations for n variable” rules of thumb sometimes fail, requiring more refined studies of solution existence and uniqueness.
- 21.
A proper derivation pathway between this axial stress/strain relationship and string curvature is fraught with subtle difficulties that we shall not review here. See [13].
- 22.
In modern usage, these are often called “constitutive principles.”
- 23.
Worse yet, the forces posited in Hooke’s law possess a character prohibited by Newton’s third law as normally construed, for they presume a natural rest configuration to which the system strives to return. For a discussion of the general problem of articulating “fundamental force laws” capable of backing up the common procedures of classical physics, see [14].
- 24.
More accurately characterized, the match situation probably invokes an “altered control variable to new equilibrium after an unspecified relaxation time” format commonly invoked within chemical and thermodynamic practice. In this essay, we have tried to sidestep detailed discussion of modeling architectures of this more complex complexion.
- 25.
For a general discussion of the inadvisability of inflating localized possibilities into possible world behemoths, see [3].
References
Goodman N (1955) Fact, fiction and forecast. Harvard University Press, Cambridge
Hadamard J (1952) Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York
Wilson M (2017) Physics avoidance and other essays. Oxford University Press, Oxford
Eddington AS (1928) Nature of the physical world. Cambridge University Press, Cambridge
Hadamard J (2010) Propagation of waves and the equations of hydrodynamics. Birkhauser, Basil
Hiddleston E (2005) Review of Woodward, making things happen. Philos Rev 114:545–547
Woodward J (2004) Making things happen. Oxford University Press, Oxford
Maudlin T (2007) The metaphysics within physics. Oxford University Press, Oxford
Paul LA, Hall N (2013) Causation: a user’s guide. Oxford University Press, Oxford
Lewis D (1973) Counterfactuals. Harvard University Press, Cambridge
Psillos S (2004) A glimpse of the secret connection: harmonizing mechanisms with counterfactuals. Perspect Sci 12(3):288–319
Psillos S (2007) Causal explanation and manipulation. In: Persson J, Ylikoski P (eds) Rethinking explanation. Springer, Dordrecht
Antman SS (1980) The equations for large vibrations of strings. Am Math Mon 87(5):359–370
Wilson M (2008) Determinism: the mystery of the missing physics. Br J Philos Sci 60(1):173–193
Lewis D (2001) Counterfactuals. Wiley-Blackwell, Oxford
Stalnaker R (1968) A theory of conditionals. In: Rescher N (ed) Studies in logical theory. Blackwell, Oxford
Correia F, Schneider B (2012) Metaphysical grounding. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Science+Business Media, LLC, part of Springer Nature
About this paper
Cite this paper
Woodward, J., Wilson, M. (2019). Counterfactuals in the Real World. In: Fillion, N., Corless, R., Kotsireas, I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9051-1_10
Download citation
DOI: https://doi.org/10.1007/978-1-4939-9051-1_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-9050-4
Online ISBN: 978-1-4939-9051-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)