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Atlantic Canada Seminar in Early Modern July 10, 2019 Halifax, Nova Scotia Tom Vinci

Leibniz’ Postulate, Planck’s Postulate and The Need for Divine in

Keywords: Leibniz, Planck, God, of Least Action, Explanation, Causal Agency

1. on Modern Physics and the Principle of Least Action Modern , in particular under the influence of the development of the of , has moved far away from Leibniz’s teleological point of view. Science has abandoned the assumption of a special, anticipating reason, and it considers each in the natural and spiritual world, at least in principle, as reducible to prior states. But still we notice a fact, particularly in the most exact science, which, at least in this context, is most surprising. Present-day physics, as far as it is theoretically organized, is completely governed by a system of spacetime differential equations which state that each process in is totally determined by the events which occur in its immediate temporal and spatial neighborhood. This entire rich system of differential equations, though they differ in detail, since they refer to mechanical, electric, magnetic, and thermal processes, is now completely contained in a single theorem, in the principle of least action. As we can see, only a short step is required to recognize in the preference for the smallest quantity of action the ruling of divine reason, and thus to discover a part of Leibniz’s teleological ordering of the universe.

1 Physics […] is inclined to view the principle of least action more as a formal and accidental curiosity than as a pillar of physical knowledge. […] On this occasion every one has to decide for himself which point of view he thinks is the basic one, and must also ask which approach eventually be more successful. Thus it has been even more surprising that this principle, originally viewed by Leibniz and Maupertuis as a mechanical theorem, was found by Hermann von Helmholtz to be valid, without any restriction, throughout the entire physics of his . Recently, David Hilbert, by employing Hamilton’s version of the theorem, has established it in Einstein’s general theory of relativity. The more complicated circumstances become, the less likely it is that the dominance of such a simple law could be a mere accident.1

2. Passages from Leibniz: Discourse on 21. If mechanical rules depended only on geometry without metaphysics, the phenomena would be entirely different. I even find that several effects of nature can be demonstrated doubly, that is, by considering first the efficient cause and then by considering the final cause, making use, for example, of God’s decree always to produce his effect by the easiest and most determinate ways, as I have shown elsewhere in accounting for the rules of catoptrics and dioptrics; I shall say more about this soon.2

22. Reconciliation of two ways of explaining things, by final causes and by efficient causes…

1 Max Planck, “Address to the German Academy on the Occasion of Leibniz Day”, Berlin, 1922, (hereafter, “Address”) in Stefan Hildebrandt and Anthony Tromba, Mathematics and Optimal Form, Scientific American Library (New York: Scientific American Books, 1985), 191-92. Hereafter, Optimal Form. The original is in Max Planck, Akademie-Anspruchen (Berlin: Akademie-Verlag, 1948), 41-48. 2 Gottfried Leibniz, Discourse on Metaphysics in R. Ariew and E. Watkins (eds.), Modern Philosophy (An Anthology of Primary Sources) (Hackett, 1988 ), 184 – 207: at 197. Hereafter, Discourse.

2 […]I find that the way of efficient causes, which is in fact deeper and in some sense more immediate and a priori, is, on the other hand, quite difficult when one comes to details… But the way of final causes is easier, and is not infrequently of use in divining important and useful which one would be a long time in seeking by the other, more physical way […] I also believe that Snell, who first discovered the rules of refraction, would have waited a long time before discovering them if he first had to find out how light is formed. But he apparently followed the method which the ancients used for catoptrics, which is in fact that of final causes. For, by seeking the easiest way to lead a ray from a given point to another point given by reflection on a given plane (assuming that this is nature’s design), they discovered the equality of angles of incidence and angles of reflection, as can be seen in a little treatise by Heliodorus of Larissa, and elsewhere.3

3. The Principle of Least Action. In its most simple formulation Action is defined as mass x distance x velocity. 4 The Principle of Least Action in the form given to it by Maupertuis (1745) is stated thus: Nature always minimizes action. A version of the Principle of Least Action can also be given for potential energy in the form of two . (1) The stable equilibrium states (that is, states of rest) of a physical system are characterized by the condition that, in such a state, the potential energy of the system is less than it would be for any possible (or virtual) close-by state of the system. (2) The equilibrium states of a physical system are the stationary states of its potential energy.

3 Leibniz, Discourse, 197. 4 Hildebrandt and Tromba, Optimal Form, 16. The two principles are taken from the same place.

3 Let’s take a simple example to explain these rules.5 Suppose we start with a valley and a ball is placed on one side. It will roll down the side until it comes to rest on the bottom. Why? One explanation has to do with a balance between the forces of gravity and the forces of resistance provided by the sides and floor of the valley. This is classical Newtonian physics. Another, which dispenses with the notion of force, relies on the fact that there is a certain point on the contour of the valley where potential energy of the whole system (ball and earth) is at a minimum, and that is on the very bottom of the valley. The bottom of the valley is the stationary point mentioned in Rule 2 and, mathematically speaking, is the point on a horizontal line that runs tangent to the bottom of the valley. This point is mathematically discoverable by applying the differential calculus to the curve representing the contour of the valley. In general, when the possible states of potential energy of a system are represented by equations, the “floor of the valley” is discovered by the differential calculus. It is Leibniz who invented this calculus in its modern formulation and put it to this use. We shall see how shortly.

4. Teleology and Minimization Principles in Optics. Catoptrics and the Minimization Principle. Leibniz identifies two approaches to physics, that “by final causes” and that “by efficient causes”. Both are represented in theories of reflection in antiquity. The earlier theory, due to Euclid around 300 BC, states that a ray of light travelling in a straight line from a point A that is reflected from a surface at point B is reflected away from B at the same angle with which it strikes that point.

5 This example and these rules are given in Hildebrandt and Trombo, Optimal Form, p. 84-85.

4 This principle can be used not only to explain why rays of light behave as they do but can be used to predict the path a ray of light will take prior to its reflection. It is because the events governed by this principle proceed forward in time that we can say that part of the explanation of the path the light takes when it leaves point B is the angle it takes when it strikes point B. Moreover, since this principle involves only the of and geometry, it counts as a mechanistic principle.6 We shall call a physical principle of this sort “causal/mechanistic” and the physics in which it occurs “forward-looking physics.”

6 The term “mechanism” in connection with physics is sometimes used to describe a physics of in motion that contrasts with “dynamical”: the latter employs a metaphysically robust notion of force, the former does not. That is the use here. However, Leibniz also uses the term “mechanistic” to describe his own physics where it describes the principles governing efficient causation. (See Gottfried Leibniz, Tentamen Anagogicum (in English) in Leroy Loemker, Gottfried Wilhelm Leibniz, Philosophical Papers and Letters (Kluwer Academic, 1989), 477- 489: at 478. Hereafter, “Tentamen”. The original, French text is in Immanuel Gerhardt Die Philosophischen Schriften von Gottfried Willhelm Leibniz, Vol. VII (Hildesheim, 1961), 270 -279.) This, of course, is a dynamical physics. When Leibniz wishes to talk about physical principles that do not involve forces, he calls them “geometrical principles.”( ibid.). Henceforth, I will follow Leibniz’s usage.

5 The later theory, due to Heron around 100 AD, states that a ray of light starting at point A, striking one surface and then reflected to another surface at point C, takes the shortest distance between points A and C. In the language of natural (Aristotelian) teleology we would say that it seeks the shortest distance between points A and C and the reflecting surface. This principle differs from the causal-mechanistic principle in that it uses teleological concepts (“seeks”), does not posit that light travels in a straight line and does not explain the behavior of the path of light in terms of the size of angle the ray takes when striking or leaving the surface. The only notion involved is the minimization of distance traveled: all other properties of the ray, its’ straight-line path, where it strikes the mirror and the angle of approach and departure from that point, are derived from this one principle. When expressed in teleological language, I shall call principles of this sort “teleological-minimization principles.” Notice that the statement of this principle requires that we are first given point C, the place where the ray of light eventually ends up. So I will call a physics having this in general (whether expressed in teleological language or not) a “backward-looking physics”; in this case it is a backward- looking teleological physics.

5. What are “final causes” for Leibniz? Before proceeding, I need to clarify the sense in which “final causes” are, and are not, teleological for Leibniz. In the passages quoted from Leibniz, he refers to “final causes.” It is tempting to take from these references that Leibniz is advocating a theory of natural teleology in the Aristotelian sense adumbrated above in which our ray of light seeks the shortest distance between two points. Indeed, Leibniz uses the language of natural teleology in an early work on physics, “Metaphysical Definitions and Reflections” (c. 1678). The passage is: “nature, proposing some end to itself, chooses the optimal means.”7

7 Quoted in Jeff McDonough, “Leibniz’s Two Realms Revisited”, 42:4 (2008), 673 – 696: at 677. Hereafter “Two Realms”.

6 McDonough there defends the that optimality explanations, in this case minimizational explanations, for Leibniz exhibit natural teleology: “Leibniz suggests that we may explain a ray of light’s passing through point B rather than point M by appeal to the end of getting to C by the quickest or shortest path possible.” It is important to note, however, that, for McDonough, natural teleology in Leibniz is ‘thinner’ than it is in .8 For Aristotle “final causes are immanent in nature – they are the immediate result of robust formal natures that are intrinsically directed towards their own appropriate ends,” but Leibniz, in the case of the optimality laws, treats the finality of final causes as “grounded immediately only in the structure of the laws of nature.” What makes a law “teleological” for McDonough is that it is finalistic in the kind of explanation that it affords, that is, it “attempts to explain a behavior or event by appealing to an outcome or consequence of that behavior or event.”9 (my emphasis) “As a consequence, Leibniz’s conception of teleology is not directly tied to any metaphysical picture of goal-directed strivings…”, that is, Leibniz’s conception of natural teleology, to the extent to which he has one, is not the thick conception of teleology. On this central point, I am in agreement with McDonough. Henceforth, when I speak of “optimality laws” I mean laws that are formulated in non-teleological language. Thus the optimality law for the path taken by a ray of light says that between start and end points, A and B, a ray of light travels the shortest distance. There is no mention of “seeking” or other teleological language here. On the other hand, when I do use terms for teleology, I will understand them in the “thick,” Aristotelian sense. There is, however, a point on which we are in disagreement. While I will not make an issue here of whether the finality of optimality laws in McDonough’s sense warrants calling them ‘teleological,’ I will make an issue of whether they afford, for Leibniz, explanations. I argue below, in the next two

8 The passages quoted below are all from Jeff McDonough, “Leibniz on Natural Teleology and the Law of Optics” Philosophy and Phenomenological Research, LXXVII #3 (May), 505-44: at 520, except where otherwise indicated.) Hereafter, “Natural Teleology”. 9 McDonough, “Natural Teleology,” 505.

7 sections, that they do not. As they occur in the natural world “final cause” principles are merely observed regularities not expressive of in-world causal or explanatory efficacy of any kind, something that I defend in the next two sections.

6. Leibniz’ Postulate In the passage quoted above from Section 22 of the Discourse, Leibniz has said that the “final-cause” principles have heuristic in helping scientists to uncover the underlying mechanical principles, e.g. the least-time principle in Fermat’s development of the mechanistic law of sines for refraction. Sometimes a principle can have heuristic value when it is false, but in this case the principles in question are true: the path followed by a refracted beam of light really does take the least-time from beginning to end of all possible paths through the two materials. Reading “minimization principles” for “final cause” principles in this context, Leibniz draws a contrast between minimization principles and mechanistic principles by saying that the latter are “deeper” and “more immediate.”10 What is the contrast? I suggest that it is an explanatory contrast: mechanistic principles are explanatory in the physical order, whereas minimization principles are not. The argument begins with some Leibnizean metaphysics: a non-Humean account of explanation that explains law-like regularities by means of metaphysically robust natures of the things being explained.

It is not enough to say that God made a general law; for besides this decree there must be a natural means of executing it; that is, it is necessary that what happens can be explained through the nature that God gives to things.”11

10 See passage quoted from Discourse, #22, at the outset. 11 (Letter to Beauval (1697): quoted in McDonough, “Two Realms,” 682.

8 Leibniz is here denying that laws stand on their own: “there must be a natural means of executing it”. He goes on to explain this idea with another: “that what happens can be explained through the nature that God gives to things.” Things lying in the nature of things that explain other things are causal agents, so Leibniz’s position appears to be that causal agents inherent in the nature of things are fundamental explainers of all particular events in the physical world, including those that fall under general laws, and that general laws are not. Because Leibniz characterizes laws as “decrees” from God, it is not entirely clear how his conception of “law” translates into worldly phenomena, but I shall assume that it does so thus: laws-in-the-world are regularities that possess nomic (physical) necessity. However, it is not quite clear to me whether, when there is an appropriate underlying means of execution, Leibniz allows that one can properly speak of laws as such as explaining why things occur (nomic explanation) or whether explanation is properly confined only to the underlying means of execution (causal explanation). I will take it that the former is correct. In this case, Leibniz’ position on nomic explanation is that a necessary condition for a law to be explanatory is that a causal agency underlies the law; and a causal agency underlies a law when the causal agency (causally) explains why the law itself is a law. This is Leibniz’s position on nomic explanation. This makes for an anti-Humean account of causality, and an anti- Covering-Law model of explanation. Taking his anti-Humean positions on causality and on nomic explanation together, they constitute what I will call “Leibniz’s postulate on causality and explanation”, or Leibniz’s Postulate for short. In the case of the laws of motion, the causal agency is force. But what of the minimization principles? Are they explanatory and, if so, what is their underlying causal agency? I discuss this question in the next section.

7. The Explanatory Status of the Minimization principles in Leibniz’ Physics

9 One possibility (Option 1) for the causal agency underlying the minimization principles is that the latter count as God’s decrees, where the “natural means of executing” them are causal agencies, the very same causal agencies that underlie the realm of efficient causation, namely, forces.12 Another possibility (Option 2) is that the minimization principles do not have “a means of execution”: they simply provide a true description of a regularity to be observed in nature without (in-world13) explanatory force. In this case the between the regularities explained mechanistically and the minimization- principle regularities depends solely on mathematical inter-derivability of the minimization principles and the mechanistic laws of motion,14 rather than on some common underlying causal agency directly. In the passage that we quoted from Discourse on Metaphysics, Section 21, (1686) at the outset, Leibniz uses the language of God’s “decrees” in connection with minimization principles in optics. This supports Option 1. But things may have changed by the time Leibniz writes Tentamen Anagogicum (1697). Here Leibniz replaces his earlier commitment to minimization principles in optics with a new principle, the principle that light takes what he calls “the most determinate path.” He says that the latter is an “architectonic” principle,”15 so Leibniz’s doctrine about architectonic explanations relates to the most- determinate-path principle (MDPP) rather than to minimization principles. However, I will not make an issue of the distinction here, calling them generally “optimization principles.” 16

12 Thanks to … 13 The qualification “in-world” is important. Leibniz accepts a distinction between physical and metaphysical levels of explanation, both real. (See note 25: re Antognazza.) I am saying that optimization-laws do not provide physical explanations (in-world explanations) not that they fail to provide real explanations – they do, but in the form of principles rationally applied by God in selecting the best possible world at the metaphysical level. 14 In Tentamen Anagogicum Leibniz shows how the derivation is to be carried out in one particular case. Tentamen, 483; G VII, 277. I follow McDonough’s interpretation, McDonough 2009, 509-11. 15 “This principle of nature, that it acts in the most determinate ways which we may use, is purely architectonic in fact …” (Leibniz, Tentamen, 484.) 16 I am treating the MDPP as an “optimality principle,” one of several that God uses in selecting the best possible world, to streamline my account here. However, I argue elsewhere that this is not in fact true, but that the MDPP is a physicalized version of the Principle of Sufficient Reason. Both the MDPP and the PSR count as architectonic principles, the former derivately, the latter primarily. See

10 Leibniz also says that the MDPP is “anagogical,” that is, it is something “that leads us to the supreme cause.”17 Leads us in what sense? One possibility is that it means “gives us the basic proof” of the of God, whose existence is otherwise epistemically problematic. Another is that, while God’s existence is not epistemically problematic, it is epistemically problematic which of the principles in nature are God’s special principles. What would be special about these principles is that they are the ones God uses directly to choose the actual world, that is, that they are the architectonic principles. The former alternative describes an approach that is clearly not Leibnz’, so the latter is the one that I adopt here.18 On this alternative, we know that the most determinate-path principles are God’s special principles in selecting the actual world because the most determinate path principle is a kind of physical optimization, and physical optimization is a kind of , so Leibniz figures that these are the principles that a being seeking perfection in His would use as the primary basis for choosing the actual world. Leibniz contrasts the status of architectonic principles with mechanistic principles in an important passage in which he distinguishes between “two kingdoms”: “… I usually say that there are, so to speak, two kingdoms in corporeal nature […] the realm of power, according to which everything can be explained mechanistically by efficient causes, and the realm of wisdom, according to which everything can be explained architectonically, so to speak, or by final causes…”19 Since optimization principles are architectonic principles and optimization principles are physical, not metaphysical principles, physical principles themselves divide into two realms: the mechanistic realm and the realm of God’s special principles where explanation proceeds by “final causes.” Because of the identification of optimization principles as architectonic and the other paper. 17 (Tentamen, 484-85, n. 2). 18 For an alternative treatment of architectonic principles in Leibniz see Duscheneau 1993, 374. He understands them in a methodological rather than a metaphysical sense. (Also see Miles 1993, 6.) For an application of the methodological interpretation to the case of the physics of light, see Duchesneau, 1993, 269-71. 19 Leibniz, Tentamen, 478-79.

11 anagogical in Tentamen Anagogicum and despite the turn to substantial forms in physics for Leibniz beginning around 1678,20 it seems that the kind of “final cause” at issue in this work must be divine purposive agency rather than immanent-in-nature , the “thick telos” of .21 In this respect the doctrine of “final causes” in physics has, perhaps, a different than is the case in the corresponding passage from Section 21 of the Discourse on Metaphysics: there, it would be natural to take “final cause” in the latter sense, rather than in the former, because of the turn to substantial forms. Now to the key question: Do optimization laws have an underlying causal agency in nature? In the passage where Leibniz states his non-Humean account of explanation (quoted above), he refers to the causal ground as a “natural means of executing” the law. This may seem to be consistent with an affirmative answer, viz., Option 1: natural forces are the natural means by which all physical laws, including optimization principles, are executed. In the latter case the execution is indirect: the mechanistic laws are directly executed, the optimization principles are made true (“executed”) by the fact they are entailed by the mechanistic laws.22 Notice, however, that the laws in question are understood by Leibniz as “decrees”, laws that God makes. I have just argued that architectonic principles are not laws of this kind: they are principles that God uses, not makes, and he uses them to choose the best possible world. In Discourse of Metaphysics, Section 5, Leibniz tells us that laws are the means God uses in creating the best world. How

20 According to Garber: “But starting in 1678 or 1679, Leibniz begins to articulate a new doctrine: even though everything is explicable mechanically, the foundations of mechanical philosophy require us to appeal to or form.” D. Garber, Leibniz: Body, Substance, Monad (Oxford: Oxford University Press, 2009): 48-49. Quoted in Antognazaa, 28. For some critical discussion of Garber see Antognazza, 29 ff. 21 Antognazza notes a shift in Leibniz’s thinking around 1670: “In these earlier texts [Confessio Naturae], in the case of non-rational , Leibniz interpreted the “incorporeal principle” or “principle of activity” needed by bodies in terms of a transcendent (that is, God) rather than as a principle of action immanent in bodies.” (Antognazza, 29) I am suggesting that, with the characterization of the MDPP in Tentatment Anagogicum as anagogical, Leibniz is returning to the spirit of the earlier doctrine. 22 This is …. view, conveyed in correspondence.

12 does he accomplish this? I surmise that starting with the optimization laws23 and in light of the inter-deducibility of the mechanistic and optimization laws, God “reverse engineers” the kind of mechanistic laws that would be needed to entail the optimization laws, “decrees” them as natural laws, then “executes” them by creating matter with a causal agency inherent in its nature (forces) appropriate for instantiating the laws. In this way mechanistic laws would be the means for achieving God’s ends, and mechanistic causal agencies (forces) would be the causal agencies that execute these laws. But since the optimization principles are architectonic principles, and the latter are are not decrees, they do not require a means of executing them; hence, a fortiori, they do not have mechanistic causal agencies as a means of executing them.; hence, by Leibniz’Postulate, they lack nomic explanatory power. QED.24 Further to the case that the optimization principles do not depend on mechanistic causal agencies: were they to do so, the kingdom of final causes would have an essential dependence on the causal/mechanistic kingdom, something Leibniz clearly does not intend. So Option 2 is preferable.

23 In choosing the best world, Leibniz’ God uses a number of principles for a number of ends, not only ones that are in the realm of physics. Clearly, there is a task for the interpreter of Leibniz to show how the use of the physical optimization principles can be integrated into these more general principles and ends, but I shall not undertake this here. Hirschmann undertakes part of this task in Hirschmann, 1988, pp. 151 – 56.

24 There are a number of other approaches in the recent literature to understanding Leibniz’ “two realms” and their relationship to one another in the case of the physics of light. It might be useful to give a comparison of my position to the following representative alternatives. McDonough 2008. McDonough takes the optimal-form explanations and the mechanistic/efficient-cause explanations to be both operating in the world and operating equipotently. (McDonough 2008, 680.) Hirschman 1998. Hirschmann takes there to be only one law operating: the optimality law understood as an architectonic principle. (Hirschmann, 158.) Duschesneau 1993. Duschesneau treats the mechanistic/ efficient-cause explanations as the only source of explanation in the physical world for Leibniz, the optimality explanations operating only at the level of mathematical descriptions, not at the level of best-possible-world-selection principles employed metaphysically by God nor at the level of explanatorily efficacious natural laws. (Duschesneau, 1993: 270-72.) A fourth approach, not represented in the literature as far as I know, is that both optimality principles and mechanistic/efficient-cause principles operate as explanations equipotently in overall, the latter with an application in the natural world, the former with a divine application to possible worlds, both expressing explanations but of different kinds. However, optimality principles are realized in the physical world only as observational regularities and not as laws with explanatory force, either mechanistic or immanent-in-nature teleological. This is the position I defend here.

13 What effect does a consideration of Leibniz’s Postulate have on our account of architectonic principles as divinely-applied rational principles God uses to select the best possible world? My answer is that it leaves it essentially unchanged. For Leibniz, architectonic principles understood as physical regularities are nomic regularities that sustain counterfactual inference but lack the power to explain events in the natural world because they lack an underlying causal agency inherent in the natural world. The architectonic principles understood as principles rationally applied by a divine reason to choose the (physically) optimal world do explain why the aforementioned regularities are as they are. They explain this because God has employed these principles to select and actualize the world in which these regularities hold. This is a genuine causal explanation grounded in the nature of God, though, because it falls in the metaphysical realm, it is not part of Leibnizean physics.25

8. The Philosophical Case for a General Version of Leibniz’ Postulate My goal in this paper is primarily philosophical: to show the need for divine reason in Physics by means of valid arguments based on the philosophical acceptability of certain premises. These premises will include, from Planck, one presupposing that optimization principles have explanatory power in the physical world (Planck’s Postulate) and, from me, one asserting a version of Leibniz’ Postulate. Is a version of Leibniz’ postulate true? I have been discussing this question in terms of Leibniz’ conception of laws understood as physical necessities in the natural realm or as decrees from God in the metaphysical realm. Both of these notions raise considerable difficulties, which it is unnecessary for my purposes to dispose of here. So I will start with the simple notion of a regularity in nature and ask whether it always requires an underlying causal agency to be explanatory, when it is explanatory. My answer is going to be Yes. This is not precisely Leibniz’s Postulate as stated

25 I follow Antognazza here in seeing a clear distinction in Leibniz between metaphysics and physics. M. Antognazza , 2017.

14 above, but is a close cousin. But when is a regularity “explanatory”? The short answer is “When it explains why something happens?” Since any position that depends on a version of Leibniz’ Postulate distinguishes between causal-types and regularity-types of explanation, if the latter is allowed at all, the account of regularity-type explanations is going to have to avoid a direct reliance on causality. To avoid this, and to secure the most liberal notion of regularity- explanation possible, I am going to rely on the notion of statistical relevance. Roughly, on this account of explanation, when E is a statistically relevant factor for X then E provides an explanation for why X happens – a statistical- relevance explanation. The fact that a person has syphilis explains (in this way) why a person acquires paresis because having syphilis is statistically relevant in an appropriate reference class to having paresis.26 Statistical relevance itself is a kind of regularity – it involves a correlation between E and X – and that is where the connection between regularity and explanation comes into this story. The connection between statistical relevance and statistical explanation has been around for quite some time and is not new. It has been discussed at length by, among others, Salmon in …27 I refer the reader to that work for the account of SR explanation that I rely on. What is new (as far as I know) is the contention that even statistical-relevance explanations conform to a version of Leibniz’ Postulate, The General Version of Version of Leibniz’ Postulate, as I will call it. Demonstrating this is the burden of the present section. First, let me state that I don’t think that all explanations have to be causal: there are also explanations invoking the laws of chance (stochastic explanations).28 But I do not think that there is an intermediate case: explanations that are both non-causal and non-stochastic. Here’s a experiment to show this. Suppose that we have three rooms that are causally isolated from one another: no transmission of can occur between them. In each room a coin is to be flipped. What is the probability that the result will be three heads? I

26 The example is from Van Fraassen 1980. 27 Reference to Salmon, 1974. 28 Cite Jefferies

15 suggest that it will be 1/8. This is because we think that the conjunction of head- turns up in room 1 and head turns up in room 2 and head turns up in room 3 is calculated by the laws of chance. So we have the following conditional statement: C1 “If there is no causal connection between the events in rooms 1 – 3, then the conjunction of events is governed by the laws of chance.” Now take the contrapositive of this: C2: “If the conjunction of events is not governed by the laws of chance then there is a causal connection between the events.” So there is no middle ground. I will call this The No Middle-Ground Principle.

We can determine empirically whether the laws of chance are, or are not, operating with a certain conjunction of events by applying the rule: If events e1 and e2 are statistically (probabilistically) independent then pr(e1 and e2) = pr(e1) x pr(e2). (This is a theorem of the Probability Calculus.) If the consequent fails to hold, then the events are correlated, i.e. not governed by the laws of chance. The notion of statistical relevance at issue in the definition of statistical-relevance explanation just is the idea that the events are not statistically independent. That is how the notion of statistical independence connects with the notion of statistical explanation and thus with the General Version of Leibniz’ Postulate. The latter says that if two events, w1 and e2 are not statistically independent, then there is causal agency in the background. If we now take this theorem together with C2 we reach the conclusion that causes have to be operating behind the scenes.

For example, suppose that we record the occurrences of thunder and lightening over a summer with check marks in three columns: Column A lists occurrences of thunder at time t, column B lists occurrences of lightening at time t, column C lists cases where they co-occur at time t. We now count check marks to determine relative frequency: if the relative frequency of check marks in column C = the relative frequency of check-marks in column A the relative frequency in column B then we have for statistical

16 independence.29 If we find that that the does not hold, then we have evidence against independence. If we do not have independence then, from The No Middle-Ground Principle, we infer the operations of a causal agency involving thunder and lightening.

For an example with numbers, if we list 1000 occurrences in each of the three columns, and in column A (thunder-occurrences) we have 40 check marks, for a relative frequency of .04, and in column B (lightening occurrences) we have 50 checkmarks, for a relative frequency of .05 and in column C we have 2 checkmarks, for a relative frequency of .002, we now compare the last figure with that obtained by multiplying the previous two numbers (.04 x .05). If we find that these are identical, as we do here, then we have evidence that the events are independent. However, suppose that for the same data in columns A and B we have for column C, 30 check marks, for a relative frequency of .03. Clearly this number is not identical with relative frequencies in column A x the relative frequencies in column B (.04 x .05 = .002). In this case, the data show that thunder and lightening are highly correlated (though not 100%), hence not independent. From this and The No Middle-Ground Principle we infer the operations of causal agency involving thunder and lightening.

The General Version of Leibniz’s Postulate and The No Middle-Ground Principle are two key premises in the case that I now propose to make for divine reason in physics. The third comes from Planck: the Principle of Least Action is fundamental in physics.

9. Planck and the Fundamentality of the Principle of Least Action in Physics. In the Address, Planck finds in Leibniz’s thought an appeal to divine reason underlying the Least-Action approach to physics, an appeal which he endorses. Although there is considerable controversy about whether Planck is

29 The evidence is a relative frequency in a sample of data. To reach the relative frequency holding in the whole population, needed for the assertion of probabilities, further steps are needed.

17 right to suppose that Leibniz himself endorses the Principle of Least Action, I will not make an issue here of the historical accuracy of this supposition.30 I will, however, suppose that for Planck the appeal to divine reason by Leibniz should be understood as an appeal to a rational agency that can apply optimality principles, including those expressive of physical optimality, to a of possible worlds to determine a unique world that satisfies the principles. I will call this the “agent-selector” conception of divine reason. The primary grounds that Leibniz has for positing an agent-selector for the universe are metaphysical and a priori. What grounds could Planck have? Planck argues that the Principle of Least Action is a fundamental principle of physics rather than a “curiosity” that really doesn’t fit into mainstream physics. When Planck says that the Principle of Least Action is a fundamental principle, I will take that as meaning that the Principle of Least Action is explanatorily fundamental in relation to competitor principles, in this case, the mechanistic principles. Is Planck right about this? Planck does not himself argue for this conclusion in his “Address”, but leaves it to “everyone to decide for himself which point of view he thinks is the basic one.”31 Planck is speaking to people working professionally in theoretical physics, and I am not such a person, so I will designate Planck as my authority on this. Designating someone as an authority on a topic, when one is not an authority on the topic oneself, is a fraught business. However, I take Planck to have a unique qualifying characteristic: as creator of the Quantum Revolution in physics around the turn of the twentieth century– not this part of physics or that part of physics, but physics as a whole -- his intuitive understanding of the whole of modern theoretical physics32 is unsurpassed; and intuitive understanding is the kind most 30 I discuss the question in the other paper. 31 See paragraph two of Planck’s “Address,” quoted above. 32 What I mean by “modern theoretical physics” is a physical theory in which the equations of Einstein’s General Theory of Relativity and Schro dinger’s Wave Equation are accepted as fundamental physical principles. By the time of Planck’s paper in 1922 this was the case for the former. The Wave Equation is the of Hamilton’s Equation to the quantization of physical magnitudes represented by Planck’s own Quantum Theory. (cite source) Hamilton’s Equation is one of two standard formulations of the Principle of Least Action, and was recognized as such by Planck in his address. Although Schrodinger did not formulate the Wave Equation until 1925, and Planck’s

18 needed when making determinations of what is explanatorily fundamental in a theory and what is not. From this I conclude that no one is better qualified than Planck to make this judgment. Before proceeding with the argument, I want to deal with an objection to the possibility that the set of least-action laws (optimization laws) in physics could, even in principle, be fundamental vis-à-vis the set of mechanistic laws. The objection depends on the fact that these two sets of laws are mathematically inter-derivable. If you start with a problem formulated within the least-action laws you can derive the corresponding mechanistic law, and vice versa. 33 The objection derives from the principle that if A and B are mathematically equivalent in the way in question here, then they are simply alternative formulations of the same phenomenon. In this case, of course, one could not in principle be explanatorily more fundamental than the other.

I do not accept the principle. Actual optimality phenomena constitute a relation among possible states or paths (that is what the term “variational” means in the term “variational calculus,” the modern terminology for the calculus of least-actions), whereas actual motion is a relation actual states or paths. So I see the phenomena as fundamentally different, and so too the mathematics, since the physical-application domains of the equations, that is, their ontological commitments, differ in the same way. So they are not the same phenomena. The oddity is that they coincide in the actual outcome. Now for the argument.

talk was given three years earlier, I take these facts to indicate that the Wave Equation was implicit in Planck’s physics of 1922. Consequently, and admitting that I am cheating a little bit, I take Planck’s position in the 1922 paper to be based on modern physics in this sense. (Cite source.)

33 As mentioned by Planck, Address. Proven in the modern-day version of the Principle of Least Action, the Variational Calculus, by Steven William Hawking and Werner Israel (eds.), General Relativity: An Einstein Centenary Survey, (Cambridge: Cambridge Univ. Press, 1980) pp. 746-789; and in Steven William Hawking and Gary Horowitz, G.T. Class. Quantum Grav. 13 (1996), 1487. [arXiv:gr-qc/9501014] Thanks to…. for this reference.

19 10. The Need for Divine Reason in Physics.

Suppose we start with optimality laws. They are, as mentioned before, to be understood non-teleologically. We already have it that all non-statistically- independent of events require the presence of a non-Humean causal agency that explains the patterns. (The General Version of Leibniz’s Postulate). Optimality regularities are non-statistically-independent patterns of events, so we need to find a suitable causal agency to explain the optimality laws. 34 We also have it that optimality explanations are explanatorily fundamental for the relevant domains (Planck’s Postulate) which, following Planck, I take to be Classical Physics and the General Theory of Relativity. So, true optimality explanations exist and, in accord with our assumptions, they must posit a non- Humean causal agency. But what is the agency?

Previously, I considered a possibility from McDonough that forces might be the causal agency not only for the mechanistic laws but for the optimality laws as well. The context at that time was a discussion of Leibniz’s view about this,

34 This is an oversimplification that I make for the sake of keeping the main line of advance relatively straightforward in the body of the text, but wish to address in this note. The No-Middle-Ground Principle allows that there are stochastic explanations – explanations that are governed by the laws of chance – as well as causal explanations. Couldn’t the world at the micro level be governed by the laws of chance, even if at the macro level the laws appear to be deterministic or nearly so? On certain interpretations of quantum mechanics, for example, the path-integral approach of Richard Feynman,* this is just what is asserted. In this case the general version of Leibniz’s Postulate is false, and so arguments that depend on that postulate, as mine does, will be unsound. There is, however, a position that takes a mixed view of the type of explanations involved in ultimate physics: some are stochastic, others are not. An example of such an approach is one in which the General Theory of Relativity is not reducible to Quantum Mechanics. There is a well-known project in “Quantum Gravity” that attempts the reduction, but this reduction is, at present, an aspiration rather than an established scientific fact. For a mathematical criticism of the program of Quantum Gravity see Alan Coley, “On the Action, Topology and Geometric Invariants in Quantum Gravity”, arXiv: 1503.04775v1 [gr-qc], 16 March, 2015. In case the criticism holds up, then the General Theory of Relativity remains as a major, irreducibly non-stochastic feature of the landscape of theoretical physics. In this case, The No Middle- Ground Principle pushes us to postulate an underlying causal mechanism to explain these laws. The General Theory of Relativity is canonically expressed with equations that amount to laws of motion, but Hilbert has argued (see Planck’s “Address”) that the formulation of the theory can be derived from a Least- Action version, and vice versa. Planck’s Postulate tell us that the Least-Action version is fundamental , so we will need to find an underlying causal agency that is suitable to Least-Action (Optimality) laws, including the aforementioned. I argue, below, that this can only be Divine Reason. So, if that argument is valid and the non-reducibility of the General Theory of Relativity holds up, then there is a set of premises that are both true and validly lead to this result. (*For an introduction to Feymann’s approach see Sean Carroll, The Big Picture (Dutton/penguin Random House, 2016)

20 and I presented a exegetical argument that Leibniz did not endorse this possibility. We now need a philosophical argument against this possibility.

Planck’s Postulate tells us that optimality laws are explanatorily fundamental in relation to mechanistic laws. But what does this mean in light of The General Version of Leibniz’ Postulate? This postulate says that laws, just on their own, are not explanatory: both the phenomena and the laws themselves are explained only by underlying causal agencies. When The General Version of Leibniz’ Postulate is in force, as it is now, my proposal is that the explanatory fundamentality of a law is to be explained in terms of a causal agency directly explaining the law. I will call this the “Directness Condition”. Suppose, for the sake of argument, that the underlying causal agency for the optimality laws are forces, the operation of which would be reflected in the mechanistic laws. Since the optimality laws and the mechanistic laws are inter-derivable, one could then explain why there are optimality laws by means of the mechanistic causal agency: the mechanistic agency explains the mechanistic laws and the latter then entail the optimality laws. That is indeed an explanation. However, it is an explanation via an intervening set of laws. In this case the tie to its’ explanatory- agency source would be indirect, via another set of laws, so the optimality laws would not be directly explained by the mechanistic agency. Hence, by the Directness Condition, a mechanistic causal agency does not provide the optimality laws with a positive explanatory status, hence depriving them of a fundamental positive explanatory status. This, in turn, contradicts Planck’s Postulate. So mechanistic causal agency cannot be the causal agency at work for the optimality laws. What could it be?

There are two remaining options for the causal agency in the case of the optimality laws, both teleological.35 One is a rational agent that selects the physically optimal world among the set of all possible worlds, the other is a non-

35 A third option, one positing an underlying stochastic mode of explaining the laws, is theoretically possible. I have discussed this possibility in the immediately preceding note, indicating the theoretical conditions under which it can be rejected.

21 rational agency that operates within the physical world. The agency in question must be a “finalist-agency” in McDonough’s sense: a cause that explains an effect by means of the outcome of the cause.36 Here we have two possibilities: a thick and a thin telos. A thin telos is ruled out because it is understood as a law rather than a causal agency. This leaves only the option of “thick” telos – a final-cause causal agency that is expressed in terms of something “seeking” an end-state -- as the right kind of non-rational agency. I now argue that this option is also ruled out.

I can’t undertake here a thorough discussion of teleological physics in general, so I will briefly make a few points. Consider, as an example, Aristotle’s notion of the tendency of fire to move upwards. This tendency, to be activated, requires that the fire be somehow responsive to, hence somehow detect, the direction up. In no other way could the tendency for fire to move up explain the actuality of fire moving up. But now we encounter a difficulty for a minimizing- telos to explain the path the ray of light takes from A to C via an intervening surface. The difficulty is that it seems impossible for the ray of light when at A to detect, hence to be responsive to, the “shortest distance between A and C via an intervening surface” because, to do so, it has to detect, hence be response to, point C. But this is just what it can’t do because it hasn’t arrived there yet! Now consider a different example. A cowboy is heading westward back to the ranch after a long day on the range and wants to take the shortest path home, but his horse needs watering. The nearest source of water is a river some miles to the south. (Figure 2.37)

36 McDonough “Natural Teleology”, 505. 37 The example and diagram is based on one from Hildebrandt and Tromba, Optimal Form, p. 50.

22 The cowboy can figure out what direction to take to the river by some seat-of-the-pants calculating in order to take the shortest path home via the river. This does not require that he has to be able to detect or be responsive to the location of the ranch as such, but it does require that there be a of the location of the ranch, the river and his current location. Here there is a genuinely explanatory, and genuinely teleologically-explanatory, employment of the shortest-distance property accounting for why the cowboy arrives at the ranch. But the ray of light does not possess the requisite mental- representational and calculational capabilities. Since there are only two options – the detection-of-point-C option and the mental-representation-of-point C option – and neither is possible, no thick minimizing telos can explain why the ray of light ends up at point C. At least not by itself. But perhaps we can appeal to a division of explanatory labour: the causal- mechanistic account explains why the ray arrives at point C, the thick

23 teleological-minimization telos explains why it takes the shortest path.38 But even this may be giving too much credence to thick teleological-minimization explanations. An explanation of this sort would have to say that, having started at point A and having arrived at point C via an intervening surface, the ray of light seeks the shortest distance between A and C via the intervening surface. The difficulty is that there is no time in the of the ray’s travel at which it can do so. It cannot do so before it arrives at the end point, because, as mentioned, it must somehow detect the shortest distance, something it cannot do; but it cannot do so after its arrival for this would put the effect before the cause. (The temporal priority of cause to effect is still in force even for causes that are characterized as “seekings”: if the reason I found something is that I had been seeking it, the seeking comes before the finding.) Therefore, there is no explanation couched in the language of “seeking” that explains why the ray took the path that it did, given start and end points as its boundary conditions, thus no thick teleological-minimization explanation for this. Moreover, we have previously seen that a thick teleological-minimization explanation cannot explain why the ray reaches the end point given its start point and the presence of intervening and final surfaces as initial conditions. So a thick teleological- minimization explanation can explain nothing at all about the path the ray of light takes. I have now argued that the causal-agency underlying optimality explanations cannot be a thick natural telos. So, the only available causal agency remaining that respects both The General Version of Leibniz’ Postulate and Planck’s Postulate is rational agency, that is, the agency of a divine reason applying a global measure of physical optimality to select, and then actualize, the best of all possible worlds. QED.

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