The Principle of Least Action (Chile)

The following presentation is another entry in my attempts to understand the principle of least action from a Neo-Aristotelian perspective. It was presented at First Chilean Conference on the Philosophy of Physics. In the presentation, I engage the views of Vladislav Terekhovich and Vassilis Livanios, who have both provided keen counterpoints to dispositionalist approaches to this subject. Livanios has also given a most helpful “Challenge” to the dispositionalist by outlining the resources available and the shortcomings of that ontology. The paper presents two lacunae, one of particular interest, which dispositional ontologists of the stronger, Neo-Aristotelian or Thomistic variety much attend to if global laws, conservation principles, or other similar key concepts of modern physics are to be incorporated into a broader Aristotelian-Thomistic philosophy of nature.

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The Action and Power of the Universe (Part 2)
The Principle of Least Action and Our Knowledge of Nature

John G. Brungardt
Postdoctoral Fellow, Instituto de Filosofía
Pontificia Universidad Católica de Chile

– Introduction

The purpose of this presentation is to outline an interpretation of the principle of least action (or PLA) using dispositional ontology in general and a Neo-Aristotelian approach in particular.¹ Dispositional ontology—or an ontology of powers—is opposed to at least two other philosophical schools concerned with the laws of nature: a Humean regularity approach (e.g., David Lewis) and a nomic necessitarian approach (e.g., David Armstrong). To describe it briefly, dispositionalist ontology generally maintains that the ontological “furniture of the world” must include, apart from actualities, properties that are not actualities or manifestations but are themselves “dispositions”—a property of being disposed to, prone to, potent to, or having a power to act or manifest other properties. Hence some versions view are called Neo-Aristotelian, for Aristotle claimed millennia ago that being or what exists is divided into what exists in act and what exists in potency.² Dispositions or powers are mind-independent realities that are not mere ways of speaking or psychological projections, they exhibit an order to or directedness to their manifestations, and they still exist even when their correlative manifestations do not (unbroken glass is still fragile). This dispositional ontology becomes Neo-Aristotelian when one attempts to incorporate more robust, updated claims about hylomorphism and a four-cause analysis of the natural order, including teleology. So, our question is whether this expanded tool-kit helps or hinders us when discussing physical principles like the PLA.

There is a growing discussion in the philosophical literature about whether dispositional ontology can shed any philosophical light on the PLA. This presentation will focus on two papers in this field: one by Vladislav Terekhovich, the other by Vassilis Livanios. Livanios, who is not in favor of dispositionalism, actually answers several key objections against the dispositionalist himself; however, Livanios also raises an important unanswered difficulty for the dispositionalist. Likewise, the views of Terekhovich also highlight by contrast the demands placed upon a dispositionalist interpretation.

We proceed in three parts. First, I highlight some aspects of the PLA. Second, I consider the views of Terekhovich and then those of Livanios. Finally, I outline a Neo-Aristotelian, dispositionalist ontology of the PLA, taking into consideration the objections raised against it. The PLA, if it is grounded by a disposition, is only grounded by a disposition of a far different sort than is usually considered. Our conclusion: The PLA is a global, bottom-up effect from the perspective of mathematically local, differential dynamics; it is a physically global condition for coordinated interaction when one sees the universe as composed of objects with real power for action and motion.

1. Background for the PLA

First, I highlight certain aspects of the PLA. Mathematically, the PLA is a time-integrated Lagrangian with a stationary value. The integrated Lagrangian yields the physical quantity called “action.” When used with the Euler-Lagrange equations, one can derive the equations of motion of a system. Action as a physical quantity is measured in Joule-seconds, and so a minimized or stationary value with respect to alternative motion paths represents a certain character to the use of energy in the physical system through time. This energy-character is specified by the Lagrangian. The Lagrangian is, therefore, crucial to joining the PLA (as a mathematical tool in the calculus of variations) to physical reality. Similarly, the alternative possible “histories” or motion paths must be understood by looking to this physical tie as a governing factor.

2A. Terekhovich: Leibniz or Aristotle?

We now turn to Terekhovich’s consideration of the PLA. Terekhovich attempts to ground the PLA using a Leibnizian view of modality in lieu of a “possible worlds” view of modality (on this, he and the Neo-Aristotelian dispositionalist agree), and Terekhovich uses his Leibnizian model to reinterpret the dispositionalist’s view of the PLA.³ The Leibnizian metaphysical model has two levels. The first level concerns the reality of possibilia, and the second level proposes how the possibilia become actual.

Regarding the first level, Terekhovich agrees with the dispositionalist in rejecting the reality of possible worlds and distinguishing such worlds from our world’s possibilia.⁴ He divides the unique, real world in two: the possible modality and the actual modality. The possible modality includes the totality of all “possible events and histories,” which all, as long as they are not self-contradictory and consistent with the laws of physics, “have essences but do not have existence” (i.e., they are not observed) and thus these alternate possibilities “‘occur’ simultaneously in the possible realm of our world. The actual history is naturally consistent with the physical laws of our world and occurs in the only actual realm of our world.”⁵

The second level of the model explains how the possibilia become actual; here, Terekhovich uses an analogy to Feynman’s path integral formalism: just as quantum histories with the highest probability promote an actual history, so also for all possible histories of motion those with the highest degree of essence lead to the world’s actual history.⁶ This appeal to a Leibnizian notion of a “highest degree of compossibility”⁷ as the rule for constituting the actual world from the resources of the possible modality is governed by the PLA:⁸

Accordingly, in the modal interpretation of the PLA, of the infinite set of the possible histories, only the one with the minimal action can exist as actual because it has the highest degree of essence and combines the greatest number of possibilities at the same time. It appears the more essence a possible history has, the less action there is.

On the basis of this model, Terekhovich proposes that the dispositionalist could adopt his interpretation of the PLA by thinking of the alternative histories in the possible modality as unrealized dispositions: “The dispositions of actualized histories differ by degrees of necessity in being manifested in the actual modality, and the degree of necessity can be measured by the value of the action.”⁹

However, it is unclear how his view explains rather than stipulates that the PLA is that reason due to which “the maximal number of possible histories” are combined. Indeed, what makes it possible to combine these possible histories in a “maximal” way, besides a brute-fact appeal to the PLA? That is, the possibility to combine possible histories must be a possibility in a different sense than that possessed by the alternate histories and therefore requires its own analysis. Besides, the dispositionalist typically takes a disposition’s being possible to mean that it is realizable or able to be manifested, whereas, on Terekhovich’s approach, there are infinities of so-called “possibilities” that have no such disposition (because it is not possible to combine them into a maximal number of possible histories) and, therefore, they are not “possibilities” in the same sense.

So, we are left with the following results: First, some distinction must be made in the senses of “possibility.” Second, what Terekhovich has done, through his laudable focus on the nature of possibility and actuality in a single world, is to highlight the need for dispositional ontology to clarify how dispositions and manifestations are related in the very constitution of the world’s history as a whole.

2B. Livanios: Has the World an Essence?

We now turn to Livanios’s consideration of the PLA. In his 2018 article, Livanios argues on behalf of the dispositionalist to answer three objections against its view of the PLA.¹⁰ Recall that, ultimately, Livanios is not a dispositionalist. Indeed, his third point leads to what I will call Livanios’s Challenge.¹¹

The first point concerns the difference in the modality of logical possibility and physical possibility.¹² The objector to dispositionalism maintains that the PLA “presupposes that the action of any given physical system could metaphysically have taken different values.”¹³ That is, the alternative motion paths are really possible. However, this is contrary to a dispositionalist ontology, for a dispositionalist evidently maintains that “there is only one metaphysically possible quantity of action and just one metaphysically possible sequence of states.”¹⁴ To this objection, however, Livanios replies on behalf of the dispositionalist—and rightly, to my mind—that one should distinguish between logical possibility and physical possibility. We slightly adapt his reply here: The alternative histories are possible in the logical space of our mathematical imaginations. This logical possibility of alternative histories is a necessary but not a sufficient condition for the application of the PLA and therefore for understanding the meaning of “physically possible.”

The second point concerns the priority of certain types of explanations of physical systems. The objector to dispositionalism points out that actual scientific practice uses the PLA to derive the equations of motion for a system, which equations tell us about the systems’s character. The order of explanation according to the dispositionalist, however, is in the contrary direction. The dispositions of the objects in a system give rise to the laws or equations of motion, and these are codified in general by least action principles.¹⁵ To this objection, however, Livanios replies on behalf of the dispositionalist—and rightly, in my view—that we must attend to the nature of the Lagrangian at the heart of the PLA.¹⁶ In order for the PLA to provide a physical explanation, one must have found the Lagrangian that is proper to the system being studied. Thus, the Lagrangian stipulates the properties and hence—the dispositionalist is free to claim—the physical dispositions of the system involved. Consequently, physical dispositions can still underwrite the laws of a system’s temporal evolution discovered mathematically using a least-action analysis.

These first two responses distinguishing between logical and physical possibility and the dependence of the Lagrangian upon physical dispositions go a long way to distancing dispositional ontology from Terekhovich’s Leibnizian modal view; the tools of our mathematical physics and the possibilities which they imply in our eidetic variations reveal options beyond what nature in fact exhibits and thus deems “possible” in a physical sense.¹⁷ Despite these defenses, however, Livanios notes that under certain conditions there is an “equivalence of the description of motion by differential equations of motion and by integral action principles,” and then he infers: “A potential worry is that the consequences of this equivalence render the whole debate under consideration metaphysically insignificant or redundant.”¹⁸ In what follows we will see if the dispositionalist is metaphysically otiose.

The third point Livanios discusses is whether dispositionalism is compatible with grounding both differential and integral explanations of motion.¹⁹ Here, the objector points out that dispositional ontology for mathematical physics is entirely local in its claims, i.e., local in the mathematical sense of being restricted to an infinitesimal neighborhood of points.²⁰ Dispositionalism can only ground differential equations of motion through this local ontology. The PLA, however, is a mathematically global and not a local explanation, and is therefore incompatible with dispositionalism.

Livanios proposes that the dispositionalist can meet this charge by reevaluating how the PLA is an explanandum.²¹ That is, the dispositionalist must reinterpret the PLA so it is no longer a law that flows from the object-level dispositions of physical systems in the way that differential dynamic laws usually do. Rather, the PLA should be deemed a meta-law to which other laws about physical objects must conform. How, then, would dispositional ontology provide an explanation for the PLA as a meta-law?

Here, Livanios raises what he considers the unanswered difficulty facing the dispositionalist. The most plausible way one might provide for a dispositionalist ground of the PLA as a meta-law is by trying “to show that [the PLA] ‘flows’ from the dispositional essences of the world” and thus extending the scope of dispositionalism’s application “from the object-level to the law-level.”²² Some dispositionalists take this route and propose that there exists a dispositional property called “Lagrangianism.” Lagrangianism is “of the essence of all physical systems. It is a truly universal property without which no physical object could be a member of the most general kind of substances existing in the world.”²³

Livanios argues that there are two difficulties with such a proposal. First, it is ad hoc—there is no measurable, physical property corresponding to this “Lagrangianism” and, as a sui generis property, its only role is to save face for a dispositional grounding for the PLA. Second, even if the appeal is not ad hoc, it is redundant. The reason a world-disposition like “Lagrangianism” grounding the PLA is redundant is that the Lagrangian itself already relies upon the dispositional properties of the objects composing physical systems. So, what independent role does this world-disposition of “Lagrangianism” play in our explanations over and above an individual object’s dispositions?

The difficulty for the dispositionalist, therefore, can be stated as follows. I will call it Livanios’s Challenge: How can the local origins of dispositional ontology explain a global meta-law like the PLA?

3. The Neo-Aristotelian Proposal

To motivate my dispositionalist interpretation of the PLA, I first propose some distinctions about “local” and “global” at different levels of conceptualization. I will then use these distinctions to answer Livanios’s Challenge.

First, we should briefly consider our talk of “local” and “global.” These terms exhibit a manner of systematic equivocation on three levels: the mathematical, natural-scientific, and ordinary language levels. That is, “local” and “global” are analogous terms.²⁴ The mathematical physicist might trade on similarities between these levels, but we must keep them distinct. What is true about one level of the local vs. global distinction is not necessarily true at another level.

Furthermore, recall that Livanios had reformulated the PLA as a meta-law on behalf of the dispositionalist so as to avoid a conflict with the presumed commitment to grounding differential laws of motion at a local scale first and not the PLA at a global scale. However, this implies—incorrectly—that the dispositionalist favors a sort of pointillisme as an ontology. Jeremy Butterfield defines pointillisme as “the doctrine that a physical theory’s fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there, so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts.”²⁵ However, this is not true. Dispositionalists, especially Neo-Aristotelian ones, are concerned with dispositions that are not mathematically local in this pointilliste sense (e.g., the dispositions of a living organism as a whole or in its parts are not mathematically local). More crucially, a Neo-Aristotelian analysis of physical motion cannot be pointilliste because motion as a dispositional reality is mathematically non-local. The mobile object when actually moving possesses a disposition to later places or states where it can end up. To say this in another way, the Neo-Aristotelian permits the dispositional non-locality of motion, where “non-local” is a denial of “local” in the mathematical sense. This denial that moving objects are “local” in the sense employed by mathematical physics allows the possibility that more-than-mathematically-local realities are fundamental. What is “local” or “global” in meanings that are available to the natural sciences or to a natural philosophical or metaphysical analysis can now be brought to bear.²⁶

Now, to answer Livanios’s Challenge: I first deny that a dispositionalist must prioritize a grounding ontology that is “differential” or pointilliste, that is, local in a mathematical sense. The properties involved in physical motion cannot be defined as intrinsic to points but must make reference to extrinsic properties, among which some are dispositional in nature. If we can back out of the commitment to a differential ontology, what do we put in its place? Here, the dispositionalist must say more by elaborating a dispositionalist theory of physical continua, i.e., how dispositions lead to non-pointilliste physical properties (such as velocity, momentum, or energy). However, we must admit that such a theory is currently lacking in dispositionalist literature (at least to our knowledge); this is a first promissory note.

After this denial of pointillisme, we can distinguish how we agree and disagree with Livanios. We agree that, mathematically speaking, the PLA is equivalent to differential derivations of equations of motion. In this way, there is no priority between the local and the global. However, it does not follow from this that the physical universe is itself indifferent to the local or the global in other senses; a broader notion of “local” vs. “global” might exhibit explanatory or ontological priority. The dispositionalist therefore needs a more holistic view even of dispositions that are “local” in the restricted mathematical sense.²⁷

If there is such a level of “local” and “global” that escapes what mathematical physics can articulate in differential or integral equations of motion, then at this level the PLA might enter into our overall interpretation of nature in a different way. But there does seem to be such a local and global that escapes mathematical physics, and at two levels (at least, on the dispositionalist’s view). A first level is that even purely physical properties cannot be defined in a pointilliste fashion (e.g., velocity, momentum, energy). The second level brings a higher-order demand. The scales of physics and chemistry must be compatible with the dispositions belonging to life and to mind. On this second level, the PLA might be a condition due to a higher-order demand. Here, however, this Neo-Aristotelian dispositionalist can only offer the mission of a research program (as a second promissory note). That program must bear out the following claim: A universe that exhibits the evolution of living or thinking beings as parts or members requires the behavior of physical matter to exhibit certain characteristics, such as the PLA.

At the first level—that of non-pointillistic physical objects with dispositions for motion—the PLA is a condition for the global interaction of physical objects in the universe, where “global” is meant not a mathematical sense but in the way that the natural sciences, and especially cosmology, speak of “the global” (referring to physical regions and, ultimately, the universe). As a condition at this level, the PLA co-defines the given natural dispositions for motion. To flesh this out, recall that the principle of least action is a time-integrated Lagrangian with a stationary (or a minimum) value. The Lagrangian is an object in mathematical physics, and, as a mathematical object, it must be applied to a physical situation. (This is why, by distinguishing between logical and physical possibility, the “alternatives histories” implied by the PLA’s mathematical formalism are not physical possibilities.) This application involves physical dispositions and measurements; hence, the PLA’s Lagrangian encodes the essential dispositional properties of a system naturally in motion. I call such motion “natural” because the true path—the one with least or stationary action—is the path observed to exist in nature (or, as Terekhovich would say, in the actual modality). The PLA “encodes” this motion because the Lagrangian directly represents certain mathematical relationships defined by the Lagrangian. Because of the application to background dispositions, however, the Lagrangian indirectly represents the dispositions belonging to a physical system that is capable of or disposed to motion. This physical system, however, does not exist in a Leibnizian possible modality or in a pointilliste way, but through real dispositions that are not defined locally but by what global interaction demands—by what the universe demands. This is why we qualify our proposal as Neo-Aristotelian. Natural motions are natural because they are formal parts of a coordinated order of things. By encoding a natural motion through a Lagrangian that indirectly represents dispositions, the PLA shows itself to be a condition for coordinating the interaction of physical objects in the universe.

If the universe’s essence or nature is partly defined by this coordinated physical order among physical objects disposed for various motions, we can answer Livanios’s objection that a dispositionalist ontology for the PLA is explanatorily redundant. Recall that the PLA becomes explanatorily redundant for the dispositionalist because Livanios relocates the dispositions for the PLA from the object-level to the law-level. However, a true “system” of objects, like the universe, needs its own ontological analysis.²⁸ The Neo-Aristotelian locates the universe at the object-level and not the law-level.²⁹ Furthermore, the universe is not in one category (e.g., the universe is not a substance, pace Jonathan Schaffer) but it is a transcategorical reality, a unity of order between categories (e.g., substances, their properties, and relations). Since the universe exists at the object-level but does not exist in a single category, we need not add a single-category property of “Lagrangianism” to the object level to ground the PLA through a global disposition. This allows us to answer Livanios’s objection that proposing a dispositionally grounded essence to the world is ad hoc. If there is an essence to the world, one which we can discover by gathering the empirical results of all the natural sciences into a broad philosophical view, then claiming the existence of “Lagrangianism” or some other such property of global systems is not ad hoc but warranted. However, on our view, “Lagrangianism” would then not be a disposition of the universe as such, but (to use an old scholastic term) an “extrinsic formal cause,” part of the form of the unity of order of physical things apt for motion.

– Conclusion

In conclusion, I partially admit Livanios’s critique. The PLA is not the disposition of independent objects or individual substances; however, the PLA is still a global condition that defines the dispositions of those objects insofar as they are members of a universe. This condition permits physically global coordination and interaction between its members as a system with physical or biological modes of motion, process, and development at various scales (or it would, if the Neo-Aristotelian can make good on the two promissory notes). The PLA is a global, bottom-up effect from the perspective of mathematically local, differential dynamics; it is a physically global condition for coordinated interaction when one sees the universe as composed of objects with real power for action and motion.

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