When returning to learn from the great thinkers of the past, especially with an eye for what they can contribute to our discernment of what the modern age claims as true or to be believed, a balance must be struck so that, on the one hand, the truth from prior ages is not so emaciated in the transfer that the only connection it has with its original proponent is its author’s name, which we tack on as a helpful label to remind ourselves why it is made by that association important, and, on the other hand, that those truths from bygone eras are not weighed down by cultural and scientific baggage that encumber the truth still lurking within the work’s yellowed pages. Good intentions can lead to errors on either end, and this seems especially true when history reports that a thinker is one to be revered. St. Thomas Aquinas is frequently subjected to this. For instance, a recent short essay by Paul Krause claiming that “Thomas Aquinas’ cosmology and doctrine of the soul are vitalistic” is an example of the latter extreme, despite the good intentions of the essay’s author.
Thanks to the good folks at Universidad Gabriela Mistral, and my good friend Pablo Maillet, my short extension course, a series of lectures on “God and Philosophy,” came to a successful close this week. A short description and news story from UGM can be found here. The Spanish text reads:
Yesterday saw the successful conclusion of a course given by Prof. John Brungardt, PhD Catholic University of America, which was conducted in our university during March and April. The course treated the contemporary debate over the existence of God and whether or not it is possible to demonstrate that God exists by way of modern science. The instructor, John Brungardt, with a doctorate in the philosophy of science and a current postdoctoral researcher at the Pontifical Catholic University of Chile, reviewed the principal modern theories utilized by modern atheists. Among others, he considered the Big Bang Theory in the work of astrophysicist Stephen Hawking, as well as Charles Darwin’s theory of evolution, which tend to be employed as demonstrating the non-existence of God. Dr. Brungardt explained that these theories do not support such a claim, much less in a conclusive or demonstrative way. Among those attending the course were religious, lawyers, and philosophers, complemented by the presence of students of various fields at our university.
Now that the course has finished, we are revising the text of the lectures. They will be published as a short book: Dios y la filosofía: La existencia del divino, la sabiduría de Santo Tomás y la cosmología moderna.
The long history of the Thomist revival and its various idiosyncrasies is difficult going. Part of my research focuses upon the fruits of the tradition of scholastic “cosmology,” which nowadays we call the philosophy of nature. A new page collects and makes available some resources as part of that ongoing project.
Currently available is a draft translation of the prefaces and introduction, with some notes and other items, of Petrus Hoenen’s Cosmologia (5th ed.). Of particular interest are these words from Hoenen’s exordium:
Does one not at times pity the philosopher upon whom is inflicted the duty of teaching scholastic cosmology? For—as is suitable and particularly befitting for a peripatetic—if he wishes to diligently consult the sciences (which have accomplished much through their experiments), and if (so that he might follow them) he interrogates the physicists so as to have a great number of their answers, these contradict the scholastics, originating as they do from mechanistic philosophy. However, if he neglects them, apart from the fact that he in doing so denies also Aristotle and the great scholastics, that splendid atomic theory will always be reckoned against him, whose discussion he wishes to avoid and which, in its essential parts confirmed to a remarkable degree, will remain a possession forever.
Hoenen sought to avoid what he called a “concordism” between the Thomistic tradition and the modern natural sciences. That is, “concordism,” as I understand it, is his allusion to a method of scriptural exegesis, especially when interpreting the six days of creation in Genesis, which method attempts to broker an interpretive peace between the discoveries of the sciences and the literal text of the Bible by proposing various metaphorical or extended readings of certain passages or terms. This relates to attempts to understand the perennial philosophy of nature in relation to the modern sciences when one attempts a “facile concordism” between the two (in Maritain’s words). This analogy, as near as I can tell, was first used by Paolo Gény, “Metafisica ed esperienza nella Cosmologia,” Gregorianum 1.1 (1920): 95.
Recently, I came across this gem, written by Petrus Hoenen in his Cosmologia (5th ed., 1956, p. 305). Hoenen, who obtained a Ph.D. in physics from the University of Leiden in 1912 (writing a dissertation on thermodynamics and studying under, among others, H. A. Lorentz), writes in this context against making form out to be a being, which is against the intention of Aristotle.
The “Note XVII” to which Hoenen refers is titled: “On the error of reifying material forms.” A translation of the first paragraph:
The theory of Aristotle was perfectly understood by St. Thomas; indeed, to the point that he makes use of the clearest formulations even in the most remote deductions. Aquinas seems to have been the first one who fully understood the Stagirite; after so many ages, at last someone was found equal to the talent of Aristotle’s mind, such that through his clarity we too even now can easily understand the problem of the greatest import and the one most worth of metaphysical attentiveness: how a being is able to be intrinsically mutable.
Of course, we must also remember Ralph McInerny’s converse maxim: Sine Aristotele, Thomas non esset.
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.
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
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.
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.
1 Among the many persons to whom I owe thanks for ideas and suggestions that went into this paper are, first and foremost, Timothy Kearns and Thomas McLaughlin, with whom I participated in a long-distance collaborative project on this theme. I also thank Roberto Salas, James McCaughan, Ryan Miller, Geoffrey Wollard, Marco Stango, and Andrew Seeley for comments on drafts, to James Franklin for comments on the presented version of Part One of this paper, and José Tomás Alvarado and Fr. Philip Neri Reese, O.P., for their comments on this and related ideas. Finally, I thank John O’Callaghan and Anjan Chakravarrty for their support during a research stay at the University of Notre Dame, where I began the work for this paper. This paper was produced as part of my postdoctoral research, FONDECYT Postdoctorado, Proj. No. 3170446.
2 See Aristotle, Metaphysics, IX.1, 1045b27–1046a4. For Neo-Aristotelian views on the philosophy of science and nature, consider Ruth Groff and John Greco, eds., Powers and Capacities in Philosophy: The New Aristotelianism (New York/London: Routledge, 2013) and William M. R. Simpson, Robert C. Koons, and Nicholas J. Teh, eds., Neo-Aristotelian Perspectives on Contemporary Science (New York: Routledge, 2017).
3 See Vladislav Terekhovich, “Metaphysics of the Principle of Least Action,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 62 (2017): 189–201. He claims, ibid., 194, that dispositionalism would provide the following interpretation of the PLA: “The possible history is in our world as unrealised manifestations of possibilities, but the possible history does not exist there. The observed history with the minimal action is the realised manifestation of one of the possible histories and thus it exists in the actual world.”
4 See ibid., 195; his dispositionalist interlocutor is Alexander Bird.
5 Ibid., 195.
6 Note that Terekhovich clarifies the explanatory direction of the analogy, 196: “Despite the analogy with quantum superposition, I do not reduce the metaphysically possible histories to the quantum histories. The FPI does not explain the metaphysical combination model, since the FPI is unlikely to have an independent metaphysical essence. On the contrary, the significance of the FPI lies in one of the effects of the combination model.”
7 See ibid., 197: “Only mutually compatible essences constitute the actual world. Here, I follow Leibniz, who invoked the notion of compossibility, so that «the universe is only a certain collection of compossibles, and the actual universe is the collection of all existing possibles, that is to say, those which form the richest composite»”
8 See also 196, where he offers this qualifying clarification: “Nevertheless, it is not exactly so. As we know, in the integral variational principles, a certain system’s functional (not just the action) is stationary and takes a minimal or maximal value for the actual process among all alternative possible processes. It means that the essence and the action are not exactly the same; the former is not a definition of the latter and vice versa. Rather the metaphysical interpretation of the action (and certain system’s functionals) is one of the physical measures of the essence, which consists of the necessity of each possible history to be realised in actuality.”
9 Ibid., 197.
10 See Vassilis Livanios, “Hamilton’s Principle and Dispositional Essentialism: Friends or Foes?” Journal for General Philosophy of Science 49.1 (2018): 59–71.
11 Note that our first and second point correspond to Livanios’s first and third points. We reorder them for clarity in our response to his views.
12 Ibid., 62–64.
13 Ibid., 62. Note that Livanios, in his article, speaks of “Hamilton’s Principle” as a generalization of the PLA. We do not think any philosophical problems arise by our difference in emphasis.
14 Ibid. See also Benjamin T. H. Smart and Karim P. Y. Thébault, “Dispositions and the Principle of Least Action Revisited,” Analysis 75.3 (July 2015): 388, as well as Joel Katzav, “Dispositions and the Principle of Least Action,” Analysis 64.3 (2004): 210.
15 See Livanios, “Hamilton’s Principle and Dispositional Essentialism,” 66. See Katzav, “Dispositions and the Principle of Least Action,” 211–12.
16 See Livanios, “Hamilton’s Principle and Dispositional Essentialism,” 67.
17 This amounts to a physics-mathematics secularism. That is, just as political secularism claims to eliminate philosophical disputes about ethics from public policy without any detriment, so also physico-mathematical secularism makes old metaphysical questions about the ontological difference between physical and mathematical objects a private philosophical concern, not to be mixed with public scientific practice. This secularism is noted by Richard Hassing, who expresses the issue as a question: “Are mathematical objects different in some fundamental way from physical objects?” The ancient Greeks answered “Clearly, yes,” (think of Aristotle and Plato’s famous disagreement about Forms). However, classical physics adopted a certain physico-mathematical secularism in place of an ontology, agreeing, as it were: “Let us set aside these philosophical disputes, and assume that any difference between mathematical objects and physical objects makes no difference for the conduct of our mathematical physics.” Richard F. Hassing, “Modern Turns in Mathematics and Physics,” in The Modern Turn, ed. by M. Rohlf, 60:131–82, Studies in Philosophy and the History of Philosophy (Washington, DC: Catholic University of America Press, 2017), 169–70.
18 See Livanios, “Hamilton’s Principle and Dispositional Essentialism,” 69–70.
19 Ibid., 64–66.
20 See ibid., 64. Livanios helpfully quotes Jeremy Butterfield, “Against Pointillisme about Mechanics,” The British Journal for the Philosophy of Science 57.4 (2006): 715: “An equation of motion is called ‘local in time’ if it describes the evolution of the state of the system at time t without appealing to any facts that are a finite (though maybe very small) time-interval to the past or future of t.”
21 See Livanios, “Hamilton’s Principle and Dispositional Essentialism,” 66. Livanios also notes one solution whose price is too high for the dispositionalist to pay; see ibid., 65: “In contrast with the ‘differential’ scientific explanation for which DE [dispositional essentialism] does provide a metaphysical ground, no DE-friendly metaphysical account of the [PLA]-based scientific explanation currently exists. So, the problem for the DE-ist can be solved simply by claiming that there could be no metaphysical [PLA]-based explanation.”
22 Ibid., 68.
23 See ibid. He cites Brian Ellis, “Katzav on the Limitations of Dispositionalism,” Analysis 65.1 (2005): 90–92; see also John Bigelow, Brian Ellis, and Caroline Lierse, “The World as One of a Kind: Natural Necessity and Laws of Nature,” The British Journal for the Philosophy of Science 43.3 (1992): 371–88.
24 In common speech, “local” and “global” name aspects of our environment. “Local” is what belongs to this region or neighborhood in which one lives; the “global” refers to what embraces a whole of something, just as the globe includes the terrestrial collection of all locales. From this origin the words were transferred to a mathematical usage: “local” refers to what is true in the neighborhood of a point (or “in the small” at the infinitesimal scale); “global” names what was true of the integrated whole or the entire domain available to a given function; see James Franklin, “Global and Local,” The Mathematical Intelligencer 36, no. 4 (December 1, 2014): 6. Crucially, the mathematical notion relates local to global through a background appeal to the fundamental theorem of calculus. However, there are also natural-scientific uses of “local” and “global” which lie between the looser common usage and the more precise mathematical ones. This is required for senses where one must describe a local habitat in biology or the global conditions of, well, a planet.
25 Butterfield, “Against Pointillisme about Mechanics,” 710.
26 The PLA might not be “contained” on only one level of analysis, but might be a effect or a condition due to a different level of analysis. Or, to put it another way, a “purely isolated physical system” that behaves according to the PLA does not exist in the concrete natural order as such, i.e., an isolated physical system, and thus certain features of the PLA in our isolated analysis might be a clue to what lies beyond the physical system as such. Recent work in emergence and top-down causality seems to bear out this claim. See George F. R. Ellis, How Can Physics Underlie the Mind?: Top-Down Causation in the Human Context (Berlin/Heidelberg: Springer, 2016); also, G. F. R. Ellis, D. Noble, and T. O’Connor, “Top-down Causation: An Integrating Theme within and across the Sciences?” Interface Focus 2.1 (2012): 1–3, an editorial devoted to a special issue on the topic.
27 The key Neo-Aristotelian claim here is that the forms of things, that is, the fundamental, kind-giving actualities and powers of substances, are not causes of objects in local, point-like isolation but are potentially determinable with respect to a global array of properties. That is, an ontology of substances is compatible with both the dependent emergence of and the antecedent priority of certain global properties.
28 This need is difficult to see. In W. Norris Clarke, “System: A New Category of Being?” In The Creative Retrieval of Saint Thomas Aquinas: Essays in Thomistic Philosophy, New and Old (New York: Fordham University Press, 2009), 39–47. While we do not agree with all points of his analysis, the general lesson that Clarke draws is sound. He defines “system,” 40–41, as a “unified immanent order which [links] together groups of individuals in such a way that they form a single objectively existing or recognizable order, a single intelligible network or pattern of relations forming a whole.”
29 Contrary to Livanios (on behalf of dispositionalists) and Brian Ellis. For Ellis, see “Katzav on the Limitations of Dispositionalism,” 91: “At the summit of each hierarchy, I postulate that there is a global natural kind that includes every kind of thing in the corresponding category. The global natural kind in the category of substances is that of the physical system.”
These presentations were produced as part of my postdoctoral research project.
FONDECYT Postdoctorado, Proj. Nº 3170446
The following is a presentation given at the recent meeting of the American Catholic Philosophical Association. It is a part of an ongoing project on the principle of least action, and this version will be incorporated in some manner in a longer paper, hopefully by the end of this year. Comments are most welcome.
The Action and Power of the Universe (ACPA Version)
Ubi eras quando ponebam fundamenta terrae? Indica mihi, si habes intelligentiam.
– Job 38:4
Juvat integros accedere fontes.
– Lucretius, d.R.N. I:927
The principle of least action (the PLA) was lauded by its discoverer as follows:
There is a principle truly universal, from which are derived the laws which control the movement of elastic and inelastic bodies, light, and all corporeal substances; it is that in all the changes which occur in the universe . . . that which is called the quantity ‘action’ is always the least possible amount.2
This principle has been the focal point of debates about the foundations of modern physics ever since this enunciation by Pierre Louis de Maupertuis in the 18th century and the subsequent formalization of the PLA by Euler, Lagrange, and Hamilton. This paper defends an Aristotelian-Thomistic interpretation of the PLA.3
Some physicists propose the PLA as the centerpiece for reforming the introductory physics curriculum.4 The PLA is typically forgone in such introductions due to its technical difficulty. It doesn’t deal in the more readily imaginable forces-and-particles Newtonian model, but with expressions that need higher-level undergraduate or graduate courses in mathematics so as to be fully utilized when practicing physics. This air of being the “inner sanctum” of the physics curriculum leads one to suspect that Newton is the milk, while Lagrange and Hamilton are the meat. As one physicist observes: “[Newtonian] forces are crucial to the understanding of simple problems, yes, but just as children use counting numbers while mathematicians have graduated to the use of real numbers, so we must graduate from forces to kinetic energy and to the energy ‘structure’ functions.”5 Such comments make it sound as if initiated are those who have exchanged a Newtonian analysis for a least action analysis.
Aristotle, we may recall, begins natural philosophy with the admonition to start with what is better known to us and to end with the principles, causes, and elements that are more knowable in themselves.6 Is the PLA a principle in the natural order or only in a mathematical order? Our thesis lies in between: the principle of least action, at the level of classical mechanics, captures natural motion in its mathematical formulation. If our interpretation of the least action principle permits us to defend natural motion, especially natural motion due to gravity at universal scales, this is a promising advance for articulating a contemporary Aristotelian-Thomistic philosophy of cosmology.
Analogies and Precursors
Our first approach to the PLA is an analogy given by Richard Feynman (Figure A).7
Imagine a lifeguard (L) who sees a swimmer (S) in trouble some distance away from shore. What is the most efficient path for the lifeguard to take to reach the struggling swimmer? It is not the straight line, that is, “the path with less sand,” LAS. Nor is it the path of least water, LBS. The quickest path is somewhere in between. This path of least time, LCS, is between the first two because the lifeguard traverses the given media—running on sand, swimming through water—at different rates. He economizes on transit time using the middle route. This note of “economy” is our first hint at the PLA.
Feynman bases his analogy on Pierre de Fermat’s Principle of Least Time: “Out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.”8 Again, notice economy and optimization here. From Fermat’s Principle we derive Snell’s Law (Figure B).
Snell’s Law states the relationship between the sines of the angles of a refracted beam of light and the velocity of light in each medium. Where exactly does the point of refraction occur? Fermat’s Principle permits us to find it.
As we vary the distance x, we change the point of refraction and thus we change the total travel time, T. (In our lifeguard analogy, this is analogous to changing the lifeguard’s point of entry into the water.) In the derivation, we invoke Fermat’s Principle when we set the first derivative of the expression for the total time of travel equal to zero, since this corresponds to a minimum value—that is, the most economical amount of the total time as a function of x.
From Feynman and Fermat we can draw a general lesson to prepare for the PLA. Certain processes can exhibit economizing or optimizing features. These processes arise in situations, whether belonging to art or to nature, where one seeks to “optimize” a desired result in view of some fixed, given quantity and another, variable quantity. What shape with a fixed perimeter length has the greatest area?9 Does light reflect off a mirror by taking the shortest distance?10 Do bees use hexagonal honeycomb cells so as to hold the maximum amount of honey for the least amount of wax?11 And so forth. Such optimizations are at least heuristic clues to the inner workings of nature. For more insight, we turn to the PLA itself.
The Origin Story
Our version of the PLA’s origin story tracks the growth of classical mechanics from Newton to Lagrange; we will leave out Einstein (relativity) and Feynman (quantum theory). This story traverses a path of both increasing abstractive generality and the power to model entire systems in motion using functions that describe the energy of the system. By “system” we mean a group of mobile bodies conceived of in isolation from their surroundings, whatever their surroundings may or may not be, and where “isolation” implies a lack of pertinent causal interactions with those surroundings. This story-line departs from the forces-and-particles method of a Newtonian analysis to consider the work, energy, and general coordinates of whole systems in motion all at once, without building up a system from individual particles.
Historically, this was accomplished in two steps. First, theorists conceived of the motion of a system from moment to moment using the principles of statics (that is, static equilibrium defined as net zero “virtual work”). Second, from this “moment to moment” dynamics, theorists generalized to the motion of a whole system from start to finish. Coopersmith summarizes:
The Principle of Virtual Work yields the condition for static equilibrium: it applies at one instant and then for all time (in other words, time doesn’t come into it). D’Alembert’s Principle, being a special case of the Principle of Virtual Work, also applies at just one instant, but as we’re now in the realm of dynamics the conditions do change with time and so d’Alembert’s Principle must be reapplied at the very next instant, and then again at the next instant, and so on and so on. However, what we would like is a method that frees us from the need to explicitly re-apply d’Alembert’s Principle, and, instead, enables us to mathematically track the motions continuously and over the whole time-interval of the problem.12
In other words, the Principle of Virtual Work tells us the conditions for the equilibrium or lack of motion of a system. This “virtual work” is imagined, conceptually possible work (force through a distance). One counterfactually imagines work being done on the system so as to deduce the required forces needed for its equilibrium. D’Alembert’s Principle then extends this by iterating these conditions of static equilibrium. The iteration of static equilibrium from moment to moment is a mathematical simulacra of the dynamism of motion. The method for calculating this dynamism all at once, for a whole motion, is what Lagrange, Euler, and Hamilton jointly perfected, using the principle of least action (which, in a certain variation, is also called “Hamilton’s Principle”).
Instead of imagined changes in the path of a lifeguard, or imagined changes in the path of a light ray, or the virtual work done to a system in equilibrium, the mathematics of the PLA allows us to imagine alternative paths of motion generated by key functions defining the kinetic and potential energy of the mobile system in question. More formally, the “action” that is made “least” in the PLA is a scalar quantity, the integral denoted by S (see the formula above). The amount corresponds to the “path of least time” for the lifeguard or the light ray. Action measures the whole motion, start to finish, and, since it is an integral, it is also the summation of what is true at each point of space and time along the way. Somehow, the PLA attends to both the whole motion and all of its infinitesimal parts. The units used to measure action have the dimensions energy-time, or units of Joule-seconds (J•s); or, equivalently, momentum-meters (p•m). This definition of action by energy-through-time or momentum-through-space is significant because it adds qualitative powers (energy and momentum) to the general consideration of motion only in terms of quantities (space and time).13
One obtains the action from the definite integration between a starting time and an ending time. These correspond to the points L and S in Feynman’s analogy, or Q and P in Figure B for the light ray. The integrand L is called the “Lagrangian.” It is the difference between the kinetic and potential energy of a physical system in motion: L = K – P. (In a moment we will discuss why kinetic and potential energy define the Lagrangian in classical mechanics. The Lagrangians of quantum or relativistic physics are beyond our scope here, but I think an analogous account could be given.) For the purposes of the integration, the Lagrangian is expressed using two functions at once: a function of position designated by q, and a function of velocity designated by q̇.
Now, “the calculus of variations” developed by Euler and Lagrange permits one to compare this scalar quantity of the action to all other conceptually possible paths of motion by varying the integrated functions of q and q̇. This variation in the action is symbolized by the little ‘δ’ out in front of the integral sign and corresponds to paths that are alternatives to in Feynman’s analogy or alternative paths for the light ray. In contrast to the conceptually possible alternative values of , the true path has an action (energy-through-time) with a “stationary” or “minimum” point, just as the first-derivative of the total-time path of the light ray has a zero value. Figure C illustrates some of these unrealized, “alternative” paths that do not minimize action. (The true path is right through the middle of these alternatives.) Nature’s behavior is only captured if we ignore these mathematical alternative possibilities and use the path where the action is least or stationary or minimized.14
Before we try to deepen our understanding of the PLA, let’s note two things. First, the PLA survives in realms where Newtonian conceptions don’t. Sir Arthur Eddington observes that
the law of gravitation, the laws of mechanics, and the laws of electromagnetic fields have all been summed up in a principle of least action. . . . Action is one of the two terms in pre-relativity physics which survive unmodified in a description of the absolute world. The only other survival is entropy.15
Eddington’s words also suggest that the PLA is somehow “in things,” not only in our conception of them. If so, the PLA is predicable of a naturally ordered whole or system of mobiles. “In such systems, action is minimized,” one might say, or perhaps, “Some physical systems are Lagrangian.”16 Yet the heavy dose of mathematics required to express the PLA gives us pause. Perhaps the PLA is predicable only of a certain family of equations. Consider again our derivation of Snell’s Law. The step of “setting the first derivative equal to zero” is clearly something said of the mathematics used to derive the law and not a description of the inner workings of light itself. Second, the PLA permits arguments that are not available to Newtonian particles-and-forces mechanics. Using the PLA and assuming the homogeneity and isotropy of space or the homogeneity of time, one can derive the three central conservation laws: the conservation of linear and angular momentum and the conservation of energy.17 One can even derive Newton’s Second Law of Motion!18 Of course, such derivations follow a mathematical order: the Second Law can also be used to derive the PLA! Indeed, the natural conservation of physical quantities is what yields the symmetries that we capture mathematically, and not vice-versa.19 So the real question is not about priority in a mathematical argument but priority in physical causes.20 Indeed, both of these points emphasize that the sound interpretation of the PLA must respect how nature as a cause is prior to our formulations and conceptions of nature.
Lagrangians and Energy
We must now answer two questions. First, why is the classical (i.e., non-relativistic and non-quantum) Lagrangian defined by kinetic and potential energy? At each instant of a motion conceived in the mode of mathematical physics, there are measurable kinetic and potential energies belonging to the system. So, if one integrates the mathematical expressions for the entire motion from start to finish, it stands to reason that one would obtain two sorts of components in the entire integral, one for kinetic and one for potential energy. In actual fact this is how these defining elements of the classical Lagrangian come about. One extends this application of virtual work in statics by re-describing it dynamically—this is what D’Alembert’s Principle accomplishes. Consequently, the integration sets the equations of static equilibrium into motion, obtaining two parts to the resulting integral: one part corresponds to the kinetic energy and the other to the potential energy. It stands to reason that in order to provide some natural philosophical basis for the PLA, we must understand what this “integration” represents.
Our second question is as follows. Given below are the mathematical expressions for kinetic energy and the potential energy due to gravity.
The measure of this potential energy is equal to the product of the force of gravity (g) on a body of mass m at a height h near the earth’s surface. If one uses the PLA to examine systems involving gravitational potential energy (e.g., a falling body, a projectile, a pendulum), the Lagrangian would involve kinetic energy and an expression derived from gravitational potential energy. Note that kinetic energy always and everywhere has the same formula, while the formula for potential energy depends upon the character of the system in question. Consequently, different types of physical systems or arrangements will possess different Lagrangians, and the PLA can only be used if one is able to define that system’s Lagrangian. To repeat: the Lagrangian of a system is tied to the character of the system in question, and this character’s distinguishing feature is its potential energy. It stands to reason that in order to provide some sort of natural philosophical grounding for the PLA, we must understand not only kinetic but also and especially potential energy. This is our second question.
In order to accomplish the first task, we must keep in mind Aristotle’s treatment of the physical continuum from Physics, Book VI.21 The general lesson of this treatise is that appealing to a body’s materiality and divisibility alone will not explain motion.22 Looking at motion from the infinitesimal part, or motion’s occurrent point, or the instantaneous now of time is not enough. The full being and intelligibility of motion derive from the other three causes. Now, reflect on the very idea of a mathematical integral—our example is the action integral S. As an integral, is a continuous sum of the values of two functions (q and q̇) taken at each momentary “now” between the beginning and ending times. So, in each mathematical “now” there is a measured value of position and velocity (q and q̇) and therefore a value for kinetic and potential energy. (We discuss why and yield these energies in answer to the second question.) Notice that the mathematical formulation signifies these momentary values of energy in abstraction from or indifference to actual motion, for mathematics abstracts from motion. This is in harmony with the origin of the PLA as a generalization from and extension of the mechanics of static equilibrium. The formalism is empty of the reality which motion possesses.
What is that reality?23 The definition of motion, the actuality of what exists in potency as such, implies that motion possesses a twofold order:
An imperfect act fulfills the definition of motion both insofar as it is compared to a further act as a potency and insofar as it is compared to something imperfect as an act. Thus, motion is neither a potency existing in potency, nor is it an act existing in act, but it is an act existing in potency, (i) such that “act” designates the order of [the mobile] thing to a prior potency, and (ii) such that “existing in potency” designates its order to a further act.24
Consider a local motion. The potency of a local motion as such orders the mobile to the act of being in the terminus ad quem. The act of the local motion as such orders the mobile to a prior potency. Neither the act nor the potency signified in the definition orders the mobile to that place where the mobile is occurrently (i.e., at that very “here” and “now”). Otherwise, the mobile would not be in motion but would be at rest. Therefore, the mobile in motion exists in the infinitesimal “here” or “now” in an odd way. The twofold order provides a formal wholeness to the motion that unifies each moment of the motion to the motion’s final, yet-to-be-realized act. This twofold order obtains for the duration of the motion and integrates it ontologically. This ontological integration signified by motion’s definition is paralleled by the mathematical integration of the action integral , but they signify different unities. However, these different signified unities can be meaningfully held in parallel precisely because a body actually in motion possesses values of kinetic and potential energy at each moment, even though we do not actually stop the body’s motion to measure values. Since the Lagrangian mathematically permits us to conceive all these moments in a single grasp, we propose that the action integral mathematically models the whole of a unified, continuous motion defined in Aristotelian terms. That is, the mathematical physicist uses the integral to mentally unify the mathematical and the physical orders and “do physics.”25
Now, we must provide support our key assertion that a body actually in motion possesses values of kinetic and potential energy at each moment. This is different than the assertion that there are kinetic and potential energy values for each point of time between and , for this is a mathematical formulation. We are asking after its natural philosophical interpretation. So, we turn to our second task: What are kinetic and potential energy? Here we build on arguments of Tom McLaughlin.26 He maintains that “kinetic energy is an instance of the Thomistic notion of the act or activity of motion,” while “gravitational potential energy is an instance of Thomistic potentiality, specifically passive potentiality.”27 We examine each in turn.
The case of kinetic energy is a bit more straightforward. Even physicists describe kinetic energy as “the energy-of-motion,” and this comports with the historical genesis of the concept.28 However, what justifies the view that it tracks the actuality of Aristotelian local motion? We argue dialectically, following McLaughlin, that kinetic energy is defined in physics by the measure of how much work that a body in actual motion can accomplish.29 This argument can be amplified using the scholastic notion of quantitas virtutis or “quantity of a power,” as emphasized in a little-known work by Fr. Charles Bonaventure Crowley, OP. So, this “virtual quantity” names something that is analogously called a quantity, it does not name a potential quantity or the virtual presence of a quantity. This “quantity of a virtue” or virtual quantity is a quantification of habits, powers, or dispositions—i.e., qualities—which can involve some type of measurement of the operations and effects of those qualities.30 Now, kinetic energy manifests itself in the very occurrence of work. “Work” in this technical sense (force-through-a-distance, which is measured in Joules), is itself a virtual quantity, the measure of the ontological ability of some body to cause physical effects. Since kinetic energy is the measure of this reality insofar as a body is in motion, it makes sense that kinetic energy as a mathematical quantity represents an instance of the activity of motion.31 Furthermore, mathematical derivations of the expression for kinetic energy () appeal to the conservation of energy in the interaction of two moving bodies.32 This involves an appeal to the principle of relativity: “To obtain an absolute or invariant [i.e., conserved] quantity we must always consider one quantity relative to another.”33
However, a solution can sometimes lead to a new problem: what form shall we give to the kinetic energy of just one isolated particle? The resolution is that we are then compelled to define the kinetic energy as having the [form ]. This leads to consistency in the theoretical modeling.34
Of course, neither would Aristotelian local motion have its full reality were a solitary body to be “set in motion” in a void-universe.35 So, given that the universe is not composed of only one solitary body, the actuality of the motion of individual bodies is consistently modeled by the kinetic energy formula.
What about potential energy? In particular, we are interested in gravitational potential energy (GPE). We again expand upon McLaughlin:
The chief indication that potential energy is not an actuality but is an Aristotelian potentiality is that potential energy requires that we consider a body’s position both with respect to where the body is at some time and with respect to where it can be but is not. . . . Since a body with potential energy must include a reference to being located in a position that is attainable but unattained, potential energy must include such a position potentially. And once such a position has been attained—once the apple has fallen to the ground—it no longer has potential energy with respect to that position.36
That is, an object in the potential field of a source of gravity possesses a certain virtual quantity of potential energy due to its position, and for this reason its measurement is expressed in terms of its height. Potential energy is the energy of possible interactions, and “the [gravitational] field tells what would happen if a test [body] were brought [to a certain height],” and when released “the [body] interacts gravitationally.”37 Consequently, we can say that GPE, measured as a virtual quantity, is a natural disposition of a body for gravitational interaction due to two joint causes: its mass and its position with respect to other bodies in a gravitational field. As positional, GPE is tied to the Lagrangian’s function of q. Kinetic energy, measured as a virtual quantity, belongs to a body precisely insofar as it is in motion; therefore, it is tied to the Lagrangian function q̇.
If this is the case, then we have supported our assertion that a body actually in motion possesses kinetic and potential energy at each moment. These energies, formulated as measured virtual quantities, mathematically represent the “act” and “potency” signified in the definition of motion. The Lagrangian, defined by kinetic and potential energy, is now ready for an Aristotelian interpretation.38
Before providing one, however, we should aid our natural philosophical imaginations by considering a thought experiment proposed by Nicole Oresme in the 14th-century. This thought experiment posits “that the earth is pierced clear through and that we can see through a great hole farther and farther right up to the other end where the antipodes would be if the whole earth were inhabited.”39
Imagine dropping a stone down this hole. The stone would not “stick fast” at once in the center of the earth, but would first oscillate back and forth before coming to rest. Oresme proposes an analogy to a pendulum to support this intuition about “impetus” or “momentum.”
We can understand this more easily by taking note of something perceptible to the senses. If a heavy object b is hung on a long string and pushed forward, it begins to move backward and then forward, making several swings, until it finally rests absolutely perpendicular and as near the center as possible.40
Oresme’s thought experiment forces us to consider a distinction among the body’s inner principles of motion and how those principles relate to surrounding positions, i.e., what the environment contributes to the natural motion of a body. Similar thought experiments are still presented to students of Newtonian mechanics (Figure D). Indeed, one can prove that in this idealized case of the “Antipodal Pendulum,” the formula for the period of the stone matches the formula for the period of an everyday simple pendulum.
Now, both the simple pendulum and analogously, Oresme’s Antipodal Pendulum, have a Lagrangian:
The simple pendulum thus has a definable quantity of action. Let us contemplate the pendulum and its action in the abstract space called phase space. Figure E plots the position and velocity of a pendulum in phase space, mapping its position (θ) against its velocity (θ-dot).
The “motion” of the pendulum through phase space tracks along the arrows, in a clockwise fashion, inside the cat’s-eye shape.41 So, as the pendulum swings back and forth (or as θ goes between left and right) its velocity increases and decreases; the pendulum’s velocity is zero when θ is greatest, and its velocity is greatest when θ is zero (when the pendulum is swinging through the vertical). We can also project this two-dimensional phase space into a third dimension (see Figure F).
In this diagram, the same position (θ) and (θ-dot) are plotted on the lower plane. The vertical axis, , is the system’s total energy. The gray paraboloidal surface represents possible energies of the pendulum depending on its initial conditions S1, S2, and S3 (from lower to higher energy states); staying at these levels represents the conservation of energy. The circumference at each level is the definite integral of both and with respect to time, which is, of course, the invariant action, S, belonging to the system.
Imagine if we were to reintroduce the friction of the air or the heat generated by the pendulum’s motion. These are non-conservative forces and so energy would be lost. The clockwise “path” of the pendulum through the three dimensional phase space in Figure F would then “spiral” down and around the paraboloidal surface until it “settled” at the bottom, at rest, just as Oresme’s figmented stone eventually rests at the center of the Earth.42 We could consider other examples (for instance, the libration or “Lagrangian points” in Figure G which seem to indicate certain natural places). However, we are now ready to propose our interpretation of the PLA.
The Aristotelian-Thomistic Interpretation of the Principle of Least Action
Recall that position in a gravitational field enters into the definition of gravitational potential energy, and therefore into the definition of that system’s Lagrangian. Now, in its mathematical formalism, the PLA encodes a motion “by a continuous process of ‘instantaneous’ conversion of potential energy into kinetic energy.”43 By “encode” we mean “to signify through the symbolic mathematics of functional expressions.” In other words, the PLA’s formula directly signifies certain logical and mathematical relationships that obtain among the kinetic and potential energy values and their functions in the Lagrangian. The PLA as a mathematical formulation acts as an extrinsic formal cause.44 Physically, however, this formula can be used to indirectly signify what belongs to a mobile system in motion, for a body in motion actually possesses both kinetic and potential energy as measured virtual quantities. These values encoded by the formula at each instant can be sewn together in our thinking by seeing that these instants are aspects of a single whole, namely, the motion that we know thanks to an Aristotelian analysis. Now, our example case is gravity; we therefore argue as follows:
(1) The principle of least action is a time-integrated Lagrangian with a minimum or stationary value.
(2) Such a Lagrangian encodes the essential principles of a natural motion, measured as virtual quantities.
— Kinetic energy, measured as a virtual quantity, represents the actuality of motion.
— Potential energy, measured as a virtual quantity, represents the potentiality of the motion (e.g., due to the relationships of position in a gravitational field, or other potential fields where force is a function of distance).
— The definite integral (from beginning to end) represents the terminus a quo and terminus ad quem.
— The Lagrangian for the true path—the one with least or stationary action—is natural, for nature takes this path.
(C) Therefore, the principle of least action mathematically encodes a natural motion by integrating the Lagrangian over time.
As a corollary, this interpretation implies a conception of natural places.45 Natural places, at least for the PLA in classical mechanics, are dynamically established positions of equilibrium in the configuration of gravitating systems, which such systems possess by natural necessity.46 By “position” I mean situs, the relative arrangement of physical bodies. They are dynamically established insofar as the gravitational potential and the initial conditions of massive bodies are contingent, subject to interference, and self-influencing. Such a system cannot but have these positions (consider the libration points in Figure G), and they are natural properties insofar as they arise from the system as a whole by nature and not by art, violence, or chance.
Most importantly, they are positions of equilibrium in the configuration of the system. Here, “equilibrium” names a feature of the phase space and not the real, physical position of the system in question. It thus refers to the virtual quantities involved. As a name, “equilibrium” comes from the sense of “balancing” a weight or finding a point where a body rests due to nature, and so it is an apt analogical term. In the case at hand, bodies in orbit follow the contours of gravitational potential and thus exhibit a type of equilibrium in regard to their potential and kinetic energies. This configuration permits stable orbits based upon energy conditions and the least action principle. Using different Lagrangians, the PLA would encode other sorts of natural motions (e.g., particles in an electromagnetic field, or in relativistic or quantum dynamics). If this is true, and bearing in mind that least action figures prominently in astrophysical studies and the Einstein Field Equations that drive the standard model of Big Bang cosmology, then this interpretation of the PLA is a promising step towards a central element of the Aristotelian-Thomistic philosophy of modern cosmology.
A brief indication is possible that “natural motion” is captured by the PLA not only in classical mechanics but also in relativistic and quantum mechanics. The general notion one requires is that of a “worldline,” or the mathematical characterization of a motion through spacetime. Worldlines in classical, relativistic, and quantum mechanics are each closely related to economizing principles of action.47 We reproduce the figure from Edwin Taylor’s seminal article on the use of the PLA in the introductory physics curriculum. This figure also outlines the argument for the PLA’s centrality as a unifying principle in physics:
Taylor himself utilizes the notion of the “principle of maximal aging” to provide a definition of natural motion in the context of relativity, namely, “Natural motion is the motion that maximizes the wristwatch time between any pair of events along its path.”48 Thus, the existence of geodesics or straight-line or “free fall” worldlines are the current replacement for natural motion in relativistic physics.49
1 Among the many persons to whom I owe thanks for ideas and suggestions that went into this paper are, first and foremost, Timothy Kearns and Thomas McLaughlin, with whom I participated in a long-distance collaborative project on this theme. I also thank Roberto Salas, James McCaughan, Ryan Miller, Geoffrey Wollard, Marco Stango, and Andrew Seeley for comments on drafts, and José Tomás Alvarado and Fr. Philip Neri Reese, O.P., for their comments on this and related ideas. Finally, I thank John O’Callaghan and Anjan Chakravarrty for their support during a research stay at the University of Notre Dame, where I began the work for this paper. This paper was produced as part of my postdoctoral research, FONDECYT Postdoctorado, Proj. No. 3170446.
2 Maupertuis, quoted in Jerome Fee, “Maupertuis, and the Principle of Least Action,” The Scientific Monthly 52.6 (1941): 503.
3 Showing the sufficiency of this interpretation is beyond our scope, since accomplishing that would involve the refutation of alternative proposals.
5 Jennifer Coopersmith, The Lazy Universe: An Introduction to the Principle of Least Action (Oxford University Press, 2017), 194.
6 See Aristotle, Physics, I.1, 184a17–22.
7 See Richard P. Feynman, QED: The Strange Theory of Light and Matter (Princeton University Press, 2014), 51.
8 Richard Feynman, The Feynman Lectures on Physics, vol. 1, 26–3. See also C. R. Nave, “Fermat’s Principle,” at Hyperphysics url: <http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/Fermat.html> accessed 22–Oct–2018.
9 See Coopersmith, The Lazy Universe, 18; also, see Alberto Rojo and Anthony Bloch, The Principle of Least Action: History and Physics (Cambridge University Press, 2018), 6–12.
10 See Coopersmith, The Lazy Universe, 18; Rojo and Bloch, The Principle of Least Action, 13.
11 See Coopersmith, The Lazy Universe, 18.
12 Coopersmith, The Lazy Universe, 107.
13 We will see that this qualitative layer corresponds to virtual quantities. Also note that, since action is a scalar, it encodes the natural powers or dispositions of a motion without a preferred direction (i.e., as a vector).
14 See Fee, “Maupertuis, and the Principle of Least Action,” 498: “Whatever else action may be, it is a partnership of time and energy in which neither can be emphasized a shade more than the other.” This emphasis upon our use of mathematical tools (with their own range of “what is possible”) should help avoid problems understanding how action as a physical reality is “minimized” with respect to natural or real possibility, and not just mathematical, logical possibility.
15 Quoted in Rojo and Bloch, The Principle of Least Action, 4.
16 To paraphrase Brian Ellis, “Katzav on the Limitations of Dispositionalism,” Analysis 65.1 (January 2005): 90–92.
17 See Jozef Hanc, Slavomir Tuleja, and Martina Hancova, “Symmetries and Conservation Laws: Consequences of Noether’s Theorem,” American Journal of Physics 72.4 (2004): 428–35.
18 See Jozef Hanc, Slavomir Tuleja, and Martina Hancova, “Simple Derivation of Newtonian Mechanics from the Principle of Least Action.” American Journal of Physics 71.4 (2003): 386–91.
19 I thank James McCaughan for helpful points here. That the PLA can be easiliy conceived in the tradition of idealism or apriorism in physics is evidenced by Stöltzner’s work; see Michael Stöltzner, “Can the Principle of Least Action Be Considered a Relativized A Priori?” in Constituting Objectivity: Transcendental Perspectives on Modern Physics, ed. by M. Bitbol, P. Kerszberg, and J. Petitot, 215–27 (Springer Science & Business Media, 2009); and “Le principe de moindre action et les trois ordres de la téléologie formelle dans la Physique,” trans. by F. Marty, Archives de Philosophie 63.4 (2000): 621–55.
20 See Jozef Hanc and Edwin F. Taylor, “From Conservation of Energy to the Principle of Least Action: A Story Line,” American Journal of Physics 72.4 (2004): 521.
21 There, the Philosopher argues for various theses about motion and mobile being that follow from the mobile, motion, and time being divisible ad infinitum. Among these: The continuum is not composed of indivisible; the mobile subject must itself be divisible and therefore a body; there is no “first moment” of a motion that is “next to” its terminus a quo, nor a last moment of a motion that is not the terminus ad quem; there is no motion in an instantaneous now, nor is a moving body “in a place” in the same sense that a resting body is in a place.
22 This is especially the case in Physics VII.1 (241b24–242a15), the famous “stopping part” argument.
23 A meagre one: “[Motion] is a certain act, but the sort of act we said, difficult to know, but able to be.” Aristotle, Physics, III.2, 202a1 (Coughlin translation).
24 This twofold order is brought out by St. Thomas, In Phys., lib. 3, lect. 2, n. 3: “An imperfect act fulfills the definition of motion both insofar as it is compared to a further act as a potency and insofar as it is compared to something imperfect as an act. Thus, motion is neither a potency existing in potency, nor is it an act existing in act, but it is an act existing in potency, (i) such that “act” designates the order of [the mobile] thing to a prior potency, and (ii) such that “existing in potency” designates its order to a further act.” (My notations)
25 This is the mental act constituting the qualified unity belonging to the object of a subalternated science, in this case that of classical mechanics, which is formulated using least action. On the sense of subalternation required, see Bernard Mullahy, “Subalternation and Mathematical Physics,” Laval théologique et philosophique 2.2 (1946): 89–107.
26 Thomas J. McLaughlin, “Act, Potency, and Energy,” The Thomist: A Speculative Quarterly Review 75 (2011): 207–43. I also thank Tom for sharing a draft of an unpublished paper on the relation between formal causality and energy.
27 Ibid., 209–10.
28 Jennifer Coopersmith, “What Is Energy?” Jennifer Coopersmith’s Blog, August 26, 2017 url: <https://jennifercoopersmith.com/what-is-energy/>, accessed 24–Oct–2018. See also her Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein, Revised ed. (Oxford University Press, 2015).
29 See McLaughlin, “Act, Potency, Energy,” 220. McLaughlin provides other dialectical arguments in his unpublished paper. I add my own argument in what follows.
30 This is one of the two central contribution to the contemporary Aristotelian-Thomistic philosophy of nature and science made by Fr. Charles Bonaventure Crowley’s book outlining an Aristotelian-Thomistic philosophy of measurement. See Charles Bonaventure Crowley, O.P., Aristotelian-Thomistic Philosophy of Measure and the International System of Units (SI): Correlation of International System of Units With the Philosophy of Aristotle and St. Thomas, ed. by P. A. Redpath (Lanham, MD: University Press of America, 1996). Virtual quantity is opposed to dimensive quantity (such as magnitude or number). Quantitas virtualis is “quantity” said analogously of those qualities which are habits, powers, or dispositions. It belongs especially to those qualities related to action and passion, or agency. St. Thomas has occasion to treat of virtual quantity when he asks whether charity can increase in the soul, and answers that it can do so as to the intensity by which the subject partakes of charity. Virtual quantity can be known and measured by its effects in operation. According to Crowley, one should define key concepts of classical physics (e.g., mass, momentum, force, and energy) as virtual quantities.
31 And in this precise aspect, kinetic energy is not a potency as such, although notionally tied to potencies.
32 See Coopersmith, The Lazy Universe, 119–20, and her Appendix A6.2. She cites a proof by Maimon, as well as the argument of J. Ehlers, W. Rindler, and R. Penrose, “Energy Conservation as the Basis of Relativistic Mechanics, II,” American Journal of Physics 33.12 (1965): 995–97.
33 Ibid., 120.
34 Ibid; my emphasis and bracketed emendation for clarity. See also Coopersmith, Energy, the Subtle Concept, 342.
35 The topic has a long pedigree in the annals of scholastic natural philosophical disputation. See John Poinsot, Logica IIa, q. 19, a. 3 (ed. Reiser, I.630–32); see also Naturalis Philosophiae Ia, q. 17, a. 2 (ed. Reiser, II.365–69).
36 McLaughlin, “Act, Potency, and Energy,” 220–21.
37 See Coopersmith, “What Is Energy?” (blog post cited above). Eventually, it seems that energy must be resolved to substrata of various kinds, in particular the fields that constitute physical space. However, this is beyond our scope here.
38 Of course, this raises the question about the “genus” of these notions, namely, energy itself. Note that the mind can conceive something “through the mode of substance” or per modum substantiae. (This is the other central contribution made by Crowley in his Aristotelian-Thomistic Philosophy of Measure.) We do this in many cases. The clearest examples are in geometry or arithmetic, when we think of a triangle or a number as if they were a substance, and predicate of them various properties as accidents. The triangle has a right angle, or the number seven is prime. Physicists (e.g., Feynman), who often define energy as “that which is conserved” are conceiving energy after the manner of a substance, and thereby as the fundamental subject of their inquiry. That a conserved “something” is modern physics’s proxy for substance as a quantifiable substratum might be gleaned from Immanuel Kant’s “First Analogy” in Critique of Pure Reason, B224: “In all changes of appearances substance persists, and its quantum is neither increased nor diminished in nature” (Guyer & Wood translation). Thus, it seems more accurate to say that virtual quantities such as energy or impetus are conceived as substances and are thus fundamental concepts in modern physics. See also below, fn. 42.
39 Nicole Oresme, Du Ciel, Book II.31, translation quoted, with slight modifications, from Hall, Bert S. “The Scholastic Pendulum,” Annals of Science 35.5 (1978): 450.
41 So, as the pendulum swings back and forth (or as θ goes between left and right) its velocity increases and decreases; the pendulum’s velocity is zero when θ is greatest, and its velocity is greatest when θ is zero (when the pendulum is swinging through the vertical).
42 Given this “mixed” mathematico-physical use of abstract spaces, it becomes clear that another key background notion concerns the nature of our conception of magnitude. See Richard F. Hassing, “Thomas Aquinas on Physics VII.1 and the Aristotelian Science of the Physical Continuum,” in Nature and Scientific Method, ed. by D. O. Dahlstrom, 22:127–57; Studies in Philosophy and the History of Philosophy (Washington, DC: Catholic University of America Press, 1991), 125, n. 45, who notes that “continuous magnitude” has the following three senses: “(1) mathematical continuum, (2) physical continuum, and (3) magnitude of a body of determinate nature. The latter cannot be divided to infinity without corrupting the nature in question.” Hassing follows St. Thomas and Pierre Duhem here. Aristotle’s consideration in Book VI is not of the mathematical continuum (like triangles or spheres), nor is it about bodies of determinate natures (like human hearts, which you cannot divide to infinity without corrupting their nature). Rather, this consideration of the physical continuum is vague, indeterminate, and abstracts in an odd way from determinate natures; however, by so abstracting, it gives us demonstrative knowledge that is clearer to us. In order to progress beyond mobile natures thus indeterminately conceived, we must supply the physical continuum with qualities or powers. These powers are abstracted from when conceiving quantity mathematically, and Physics Book VI reintroduces only the most meager notion of potentiality, namely, the potential for motion in general. The fundamental insights needed to discover physical dynamics are more determinate conceptions of the dispositions and powers of bodies responsible for local motion. Keeping in mind the character of virtual quantity discussed above, it follows that the natural powers and dispositions of bodies can be subjected to measurement due to their observable effects, and thereby we can know those powers and dispositions in more detail as measured virtual quantities. These virtual quantities, or quantities of powers, help us progress beyond the indeterminately conceived physical continuum.
43 Penha Maria Cardoso Dias, “Euler’s ‘Harmony’ Between the Principles of ‘Rest’ and ‘Least Action’: The Conceptual Making of Analytical Mechanics,” Archive for History of Exact Sciences 54.1 (1999): 77.
44 I thank James McCaughan for pointing this out.
45 The insight about natural motion I owe to Ryan Miller, “Symmetry Arguments from Aristotle’s De Caelo to Noether’s First Theorem,” available at url: <https://st-andrews.academia.edu/RyanMiller> accessed 4–Nov–2018. Also, I thank Tom McLaughlin for ideas about natural place mentioned in personal communication.
46 The actual behavior of bodies differs from this aim due to interfering conditions. Orbiting bodies display various harmonic perturbations related to equilibrium points. However, the connection between potential energy gradients and natural place seems to apply even to galactic structure; see T. C. Junqueira,, J. R. D. Lépine, C. A. S. Braga, and D. A. Barros. “A New Model for Gravitational Potential Perturbations in Disks of Spiral Galaxies: An Application to Our Galaxy,” Astronomy & Astrophysics 550 (2013): A91, 10: “We have presented a new description of the spiral structure of galaxies, based on the interpretation of the arms as regions where the stellar orbits of successive radii come close together, producing large stellar densities. In other words, the arms are seen as grooves in the potential energy distribution.” It also seems necessary that natural place be a natural result of gravitational fields since most of the components of the universe either interact with those fields (dark energy and dark matter) or, of the observable matter in the universe, most of that is composed of hydrogen, which exhibits consistent natural behavior that leads to nebulae and stars, which, one might propose, are the natural effects of the universe as a whole.
47 See Edwin F. Taylor, “A Call to Action.” American Journal of Physics 71, no. 5 (April 10, 2003): 423–25.
49See Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation (Princeton, NJ: Princeton University Press, 2017), 13.
These presentations were produced as part of my postdoctoral research project.
FONDECYT Postdoctorado, Proj. Nº 3170446
The following is the abstract from an essay of mine recently submitted for review. If you would like a personal copy of the final draft, please contact me. I’d love to hear your thoughts.
This essay proposes a comprehensive blueprint for the hylomorphic foundations of cosmology. The key philosophical explananda in cosmology are those dealing with global regularities and structures, the regularity of global regularities, and the existence of the global as such. The possibility of accounting for them through alternatives to hylomorphism is briefly considered: the regularity theory of the laws of nature, nomic necessitarianism, monism, and the new Heracliteanism of Roberto Unger and Lee Smolin. These alternatives fail to meet the cosmological explananda simultaneously. Hylomorphism provides a sound philosophy of cosmology insofar as it supports a notion of cosmic essence, the unity of complex essences, and globally emergent properties. These are used to account for seven specific cosmological explananda and to resolve two problems in the philosophy of cosmology, namely, the meta-law dilemma and the uniqueness of the universe. Cosmology needs hylomorphism because it is able to ground cosmology’s efforts as a scientific inquiry. It can so this because hylomorphism philosophically accounts for changing substances and aggregates of substances, the various scales of law-governed behavior measured by the natures of those substances, and how those substances as parts relate to the universe as a whole.