Here is the final English version of a presentation which I gave in Spanish translation at the upcoming IV Congreso Internacional de Filosofía Tomista in Santiago, Chile. The Spanish translation will be posted separately.
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The Action and Power of the Universe:
Operari Sequitur Esse and the Principle of Least Action
by John G. Brungardt
FONDECYT Postdoctoral Proj. Nº 3170446
This presentation outlines a small part of my exploration of the nature of Aristotelian form and its relationships to modern scientific cosmology.1 In this presentation, we discuss the principle of least action (PLA) and its connection to the axiom that the action of a thing follows upon its being (operari sequitur esse). We will first look at some examples of the PLA and then go through a three-stage process of trying to understand it more metaphysically. We will not attend too much to the mathematical formalism that is essential to expressing the principle.2 Hopefully, despite prescinding from the mathematics, we will attend to the prior conditions of being that give physical things characteristics such as the PLA, determining them in themselves and in relationship to each other.
2. The deepest principle of nature?
The PLA is called one of the deepest principles of nature. What exactly is it? Consider an analogy given by the physicist Richard Feynman.3
A lifeguard sees a swimmer struggling out in the waves some distance along the beach from where he is. What is the most efficient path from the lifeguard station to the struggling swimmer? Would it be fastest to run along the beach first and then swim out to sea on a path perpendicular to the beach? Or is the direct path the fastest, a straight line between the lifeguard’s initial position and the endangered swimmer? The answer is neither—the fastest path is somewhere in between. This governing principle between the beginning and endpoint of a system of motion, for a given least or minimum amount of time and energy invested, is the PLA.
More formally, the “action” that is made “least” by this principle is a quantity that one obtains by the partial definite integral (taken between a starting time and an ending time) of the Lagrangian, i.e., the difference between the kinetic and potential energy of a physical system.
For all the conceptually possible paths of motion for a given system, nature uses the path where this integral sum is zero—the action is minimized.4 Nature decides to economize and save both time and energy.5 The PLA as a condition of physical systems is ubiquitous—this is an empirical discovery of modern physics.6 The PLA is a universal principle in some way. Let us turn to investigate this metaphysically in three aporetic stages.
3. Is the PLA a cause or an effect?
The first stage is to set up a question of priority: Is the PLA a cause or an effect? This can be framed in terms of the axiom operari sequitur esse: a thing’s operation follows upon the mode of its being. There is a proportion between a substance and its action or passion in the world; again, there is a proportion between the nature of a physical thing and what it does by nature. We can ask this question of causal priority with regard to the PLA.7 Is the PLA a cause (that guides physical systems) or an effect (determined by fundamental dispositions)?
Now, a Platonistic view of universals would make the PLA a cause, the governing condition of physical interaction.8 What about views that make the PLA an effect? A first possible answer is a Humean one.9 In this case, the PLA is a consequence of the contingent interactions of the “mosaic” of natural phenomena, the matters-of-fact.
If we are not inclined to swallow all the consequences of Humeanism, another answer to our question of priority is a more Aristotelian one. Perhaps the PLA is an effect of the dispositions of individual objects.10 But this is not the case according to one of the neo-Aristotelian philosophers, Brian Ellis, who says that the PLA is “of the essence of the global kind in the category of objects or substances. If this is so, then, of course, every continuing object must be Lagrangian.”11 (“Lagrangian” names the dispositional essence of the objects whose behaviors are described by the mathematical formalism of the PMA.) So, if the PLA is this sort of dispositional effect of individual objects, it is a strange one, for it constitutes part of the essence of the world, so it is also a type of formal cause.
4. The PLA: one or many?
Let’s stick with the idea that the PLA is some sort of effect that perhaps somehow emerges as a globally essential feature. Yet is the PLA a true feature because of what is true about the natures of substances? This second stage looks at the connection between a global behavior and individuals at a local scale.
First, exalt global behavior to an extreme.12 One could claim that “something is a substance if and only if it evolves by the fundamental laws”13 present in the universe. The PLA is such a law. However, if the universe is “the one and only thing that evolves by the fundamental laws,”14 then it would seem to follow that “the cosmos is the one and only substance.”15 We must find unity of being where there is unity of activity—operari sequitur esse! Now, this is not the place to combat monism. We need to find a less extreme way to understand the PLA as an effect.
Perhaps this more reserved treatment severs global behaviors from real, local dispositions.16 Anjan Chakravarrty proposes, “Why not think of the relevant dispositions regarding conservation as properties of the system itself, as opposed to properties of the things constituting or inhabiting it?”17 In this sense, a principle such as the PLA would be a property belonging to the world as a system that we can investigate empirically, for “investigation of this sort confirms that this is part of the nature of these systems; they are disposed to behave that way.”18 On this view, the property of “Lagrangianism” is not part of the essence of the global kind “the world” but is one of its properties, yet it is a property of the “the world” in itself as a simple object, while the world is composed (somehow) of the combined dispositions of various individual objects in it.
5. How is the PLA a feature of a world of substances?
Thus, according to Ellis the PLA is part of a global essence. According to Charkavarrty the PLA is part of a global effect. Both claim that the PLA is dispositional, yet neither root this disposition in individual substances. Is this possible? Could the PLA be true about the world because of the individual substances in it, things like dogs, cats, and human beings? Now, implicit in the axiom that the mode of operation follows upon the mode of being is that there could be modes of operations (plural) and thus an array of operations that follow from the being of a substance in a certain order. This order is partially captured in the order among the Thomistic categories of being.
Consider, in this regard, the 17th-century thought experiment of John Poinsot.19 How exactly is “where” part of the being of a placed thing: is it intrinsic or extrinsic? He asks: “If you were to stipulate that the whole frame of the universe be removed just as before the first creation of things, and a stone were created and set in motion, what would it acquire, if it did not acquire something wholly intrinsic?”20 To be somewhere is to be contained by a place. In a vacuum, there is apparently no containing body, but distance still exists. In the void, a hot stone could not heat, for there would be no circumscriptive body or medium to be acted upon.
John’s solution is as follows: “But if local motion were to exist in a vacuum, it would be imperfect and would acquire ‘a where’ qualifiedly and not simply speaking, but rather a mode of distance or nearness, which is a condition of [having] ‘a where’.”21 Note: a body in a vacuum would acquire a certain condition for a real where or location. This condition is having distance or nearness, presumably to points that are only mentally and not really demarcated in this hypothetical void-cosmos, like a modern coordinate system.22 Today, the conditions of being in a physical “where” are action and reaction relationships between mass and energy (on the one hand) and space-time fields (on the other); this is a modern instance of the situatedness of bodies leading to a combination dispositionally, contingently, and yet also extrinsically resulting in behavior conforming to conservation principles and the PLA.23 The ancients and medievals had ”natural place” to express this characteristic of system-level emergent effects in the universe; we have Lagrangians and the PLA. Analogously to the ancients, can we say that the extrinsic features flow from the intrinsic dispositions of things, but not reductively so?
Answer: Yes, since the formal constitution of individuals and their dispositions is double-sided. This twin aspect—both intrinsic and extrinsic determination—is a characteristic of formal causality, as St. Thomas writes: “Any creature whatsoever subsists in its own being and has a form through which it is determined as to its species and has an order to something else.”24 Consequently, the key idea is that the forms of physical objects are not merely intrinsic determinations but also extrinsically ground various relations: operari sequitur esse. The forms of individuals are—via these essential, extrinsic, determinable relations—formal parts of a cosmic whole. This comports with the Thomistic idea that the form of the universe—the natural kind of the world—is its very unity of order.25
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In review: The natures of things (including their dispositions for physical interactions) require an ordered relationship of individuals to each other. This ordered relationship of individuals to each other, by the axiom operari sequitur esse, requires an ordered relationship through an environment of possible action/passion interactions. This physical environment is encoded by the PLA. The natures of individual objects therefore interact in a way that the PLA encodes—the PLA is an essential accident of a system of natural objects (perhaps it is a mélange of the ancient categories of where, action, passion, and arrangement). However, this PLA-conforming effect is for the sake of the form of the universe, a unity of order (a harmony). This suggests that the PLA is in another respect a cause, a teleological condition for achieving an ordered cosmos. In this way, global behaviors such as the PLA flow from substances with natures that are coordinated within a whole of a certain scale, namely, the universe.
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Thanks to a question after the presentation, I would like to add the following clarification.26 We could wonder about the relationship between “system” insofar as this is used in my description of the PLA and the term “order” as used by St. Thomas. Is the PLA predicated of a logically ordered whole or a naturally ordered whole? If the PLA is predicated of a logically ordered whole, is this any give use of the PLA’s mathematical formalism and its various parts, or is it said of the class of such formulas? If the PLA is predicated of a naturally ordered whole, then is it said of the individuals of that whole (such as individual substances) or is it said of their natural whole in its very entirely, namely, the world?
At present, I am still working out my answer to this question. It seems to me that to each of the four options that I wish to reply “In a way yes, and in a way no.” The mathematical formalism of the PLA clearly pertains to the logical-mathematical order, and yet its symbolism signifies—somehow—an order in the nature of things. This is the onus probandi of a more developed version of this paper, namely, how does the mathematical formalism of the PLA signify aspects of the natures of individual things as well as the nature of the cosmic whole of which they are parts?
1 I want to thank several people whose suggestions, advice, and help made this small project possible: John P. O’Callaghan at the Jacques Maritain Center, Notre Dame; Anjan Chakravarrty; Philip Neri Reese, OP; Timothy Kearns and Thomas McLaughlin; Andrew Seeley; and José Tomás Alvarado. The investigation occurred during a research stay at Notre Dame, and was supported by FONDECYT, Postdoctoral Proj. No. 3170446.
2 My claim will be that, regardless of the mathematics used to express the principle, its interpretation must eventually resolve to non-mathematical claims that are independent of the formalism. So, we will attempt to make sense of the PLA in such a way that any additional mathematical detail or clarity would (at the most) be supportive of the position argued for here or it would (at the least) not conflict with it.
3 I borrow this analogy from Feynman. See Feynman, Richard P. QED: The Strange Theory of Light and Matter. Princeton Science Library (Princeton, NJ: Princeton University Press, 2014) 51–52. Smart, Benjamin T. H., and Karim P. Y. Thébault. “On the Metaphysics of Least Action,” (Preprint, March 25, 2013) also have a helpful analogy of a “lazy plumber” attempting to lay pipe between a well and his house that involves the least “effort” in pumping energy for a given time. Suppose we have a farmer who wants to connect an outside well to his house by a pipe so he has water for everyday use. The terrain around his house is also somewhat hilly, and he can only lay the pipe overground, not underground. He will also have to pump the water by hand for a certain amount of time every day. So, he wants to lay the pipe so that he needs to exert the least amount of effort in a given time every day when he pumps water. In order to conserve effort, it is not the case that the path the pipe must take is the direct path overground. It will depend upon the terrain between the well and his house. In this example, given the constraints of the terrain and the time available to pump water between the two endpoints (the house and well) the “least amount of effort” corresponds to what we will call the “action”.
4 It is important that these alternative paths are “conceptually” possible; they are “alternatives” in a restricted sense.
5 See Jerome Fee, “Maupertuis, and the Principle of Least Action,” The Scientific Monthly 52, no. 6 (1941): 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.”
6 The behavior of light, solids, liquids, and gases obeys this principle. The PLA applies to the behavior of quantum phenomena at the smallest scales as well as the behavior of systems of galaxies and of the universe at the largest scales. Thus, by induction, the PLA seems to qualify as a universal principle—it holds over all scales and all states of matter (that is, mass and energy).
7 Benjamin T. H. Smart and Karim P. Y. Thébault, “Dispositions and the Principle of Least Action Revisited,” Analysis 75, no. 3 (2015): 392: “Does the [PLA] guide physical systems such that action is [minimized], or do the fundamental dispositions of objects have a modal character that determines the [minimized] quantities of action inherent to all physical systems?”
8 One could conceive of the laws of nature as higher-order universals that constitute the necessary relationships between lower-order universals instantiated in the world (à la David Armstrong). Thus, laws of nature such as the PLA, Maxwell’s equations, or gravitational attraction would all be such higher-order universals. A consequence of this view is that the higher-order or more universal law (in this case, the PLA) would be more of a law and more of a cause. Perhaps it is the only law (to avoid overdetermination). If we accept arguments against Platonism (such as variations of the “Third Man”), or maintain that some sort of dispositionalism must be involved, we will be disinclined to accept this explanation of the PLA as a cause.
9 Smart and Thébault, “On the Metaphysics of Least Action,” 16: “According to Humean Supervenience, the PLA is a contingent truth that supervenes on the [spatio-temporal] mosaic of property and relation instantiations. For the Humean, the PLA is a law in virtue of the path the physical system follows being that which extremises action.”
10 Consider the dispositionalism of Brian Ellis in this regard: “Katzav on the Limitations of Dispositionalism,” Analysis 65, no. 1 (January 2005): 90–92, which is in response to Joel Katzav, “Dispositions and the Principle of Least Action.” Analysis 64, no. 3 (July 2004): 206–14. Katzav responds in the same issue: “Ellis on the Limitations of Dispositionalism.” Analysis 65, no. 1 (January 2005): 92–94. The debate is revisited by Benjamin T. H. Smart and Karim P. Y. Thébault in their “Dispositions and the Principle of Least Action Revisited,” Analysis 75, no. 3 (July 2015): 386–95.
11 Ellis, “Katzav on the Limitations of Dispositionalism,” 91–92. Ellis defends a threefold hierarchy of natural kinds: 1) “of objects or substances,” 2) “of events or processes,” and 3) “of properties or relationships.” (Ibid., 90–91.) The PLA fits into the “summit” of the first hierarchy: “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.” (Ibid., 91.)
12 This is similar to the monistic argument from nomic regularity proposed by Jonathan Schaffer, “The Action of the Whole,” Aristotelian Society Supplementary Volume 87, no. 1 (June 1, 2013): 67–87.
13 Ibid., 67.
16 See Anjan Chakravartty, “Physics, Metaphysics, Dispositions, and Symmetries – à La French,” (forthcoming). His view develops the idea of the world as a member of a natural kind proposed by 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, no. 3 (1992): 371–88.
17 Chakravartty, “Physics, Metaphysics, Dispositions, and Symmetries.”
19 CP:Logica, II, q. 19, a. 33 (p. 548–59 Vives ed.; c. 632 Reiser ed.). Recall the categories of ubi, or “where,” along with action, passion, and orientation (or arrangement, situs). Motion is intricately connected with these four categories, for the quantitative concomitants of motion are expressed in these categories and provide the substructure for the qualitatively teleological character of natural motions.
22 One might recall, in this regard, abstract problems involving the conservation of energy in modern physics such as the gravitational oscillator problem. This thought experiment involves a body falling down a tunnel through the center of the earth, emerging on the other side, and falling back through again, oscillating (in this idealization) ad infinitum.
23 This nexus of dispositionality, contingency and ‘extrinsicity’ is debated between philosophers of the caliber of French and Bird; and this network of concepts and its relation to the ontology of physical fields or spacetime as a quasi-substance would require a broader treatment than what I can now give.
24 St. Thomas, ST, Ia, q. 45, a. 7, c. “Quaelibet enim creatura subsistit in suo esse, et habet formam per quam determinatur ad speciem, et habet ordinem ad aliquid aliud.” (Leon.4.476)
25 See St. Thomas, ScG, II.39: “Forma autem universi consistit in distinctione et ordine partium eius.” (Leon.13.358:13–15); Super Sent., lib. 4 d. 48 q. 2 a. 4 arg. 5. and ad 5; “… bonum universi, quod consistit in ordine et harmonia, dignius est quam bonum alicujus naturae singularis.” De potentia, q. 6 a. 1 arg. 7: “Praeterea, bonum universi est bonum ordinis, ad quem pertinet cursus naturalium rerum.”
26 I thank Ricardo Salas Venegas for his question.
This presentation was produced as part of my postdoctoral research project.
FONDECYT – POSTDOCTORATE, Proj. No. 3170446.