The only way to avoid becoming
a metaphysician is to say nothing
E. A. Burtt.
Abstract
In this paper I argue that inconsistencies in scientific theories may arise from the type of causality relation they—tacitly or explicitly—embody. All these seemingly different causality relations can be subsumed under a general strategy developed to defeat the paradoxes which inevitably occur in our experience of the real. With respect to this, scientific theories are just a subclass of the larger class of metaphysical theories, construed as theories that attempt to explain a (part of) the world consistently. All metaphysical theories share a common structural backbone specificially designed to defeat paradoxes, their often wildly diverging ontological claims notwithstanding. This common structure shapes the procedures which govern the invention of ideas in the context of such theories, by codifying some onto-logical a priori assumptions regarding the consistency of reality into its bare conceptual framework. Causality plays a key rôle here, because it implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalising power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of ‘system’ and ‘lawful behaviour’. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. The fundamental backbone’s specific gestalt thus functions as a theory’s individual signature and paves the way to a comparative historical approach towards their study. An important part of my paper therefore explores the strong connections between paradoxes as they appear and are dealt with in ancient philosophy and their re-appearance in early modern natural philosophy and science. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results.
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Notes
For Aristotle’s works we used the edition in the Loeb Classical Library (1933).
I owe this reference to Strauss, in his paper on the Excluded Middle (1991).
Ainsi le principe de causalité n’est que le principe d’identité appliqué á l’existence des objets dans le temps (Meyerson 1932, p. 38). The rôle of conserved quantities in causal theories has become something of a hype since Phil Dowe’s paper (1992). See Kistler for a criticism (1998). None of these, however, mentions Meyerson’s pioneering work. In what follows I refer to the English translation of Meyerson’s book (1960).
As Aristotle makes very clear in the first book of the Metaphysics. The idea to separate being from non-being both on the ontological and the epistemological level as a strategy to defeat the paradoxes and inconsistencies of his predecessors, is already developed by Plato in the Theaetetus, the Sophist and the Statesman. Indeed, Plato’s philosophy marks the birth of metaphysics in every sense (Verelst 2008).
This scheme is more than just a fancy device. There is a recent branch of mathematics, category theory, which is particularly useful when dealing with this type of structural relationship. In category theory there is a specific kind of relation which captures behaviour and formal properties of structural connections between the local and the global level: adjunction. Now our claim can be summarised as follows: causation in (meta)physical theories has the formal structure of an adjoint. Which adjoint? Let an example suffice for now to make the point: if one interprets causality in terms of order relationships, then we know already that this adjunction exists as the Galois connection. To put the idea a bit more formally: if causality can be expressed by \(\leqslant \), then there exists a categorical duality expressed by a pair of adjoint functors \(Glob\) and \(Loc\) (with \(L \vdash G\)) (Borceux 1994, p. 96sq.). Galois connections abound in information theory, where \(\leqslant \) is interpreted as logically “stronger than” (see Vickers 1996, p. 134), and in quantum logic, where it translates the idea of causal power into physically “stronger than” (Piron 1976, p. 20), as well as in philosophy of physics, with Leibnizian conceptions of “branching space-time” (Belnap 1992). We claim that appropriate adjoints exist for all possible interpretations of causation featuring the structural characteristics outlined above.
The extant fragments of the pre-Socratic philosophers are available in the critical edition by Diels and Kranz (1951). I follow scholarly custom in my references to that edition.
Remark that there is no dilemma involved (the text has ‘and’ [
], not ‘or’ [
]). In fact this holds for the motion arguments as well. This was recognised by ancient commentators, e.g. Simplicius in his attestation that [In his book, in which many arguments are put forward,] he [Zeno] shows in each that stating a plurality comes down to stating a contradiction [
—Simpl., Phys., 139 (5) (cfr. DK 29B 2)]. I believe that on this analysis, it is possible to build a mathematically rigorous representation of all Zeno’s paradoxes along the lines of Lawvere’s categorical characterisation of “cohesion”, as it implies a duality between the continuous and the discrete. We shall save this interesting subject for a future paper. Cfr. (Lawvere 2007).
Quoted by M. Blay in a delicious little book on the infinite (Blay 2010, p. 40).
Haecque indefinita dicemus potius quam infinita: tum ut nomen infiniti soli Deo reservemus, quia in eo solo omni ex parte, non modo nullos limites agnoscimus, sed etiam positive nullos esse intelligimus; tum etiam, quia non eodem modo positive intelligimus alias res aliqua ex parte limitibus carere, sed negative tantum earum limites, si quos habeant, invenire a nobis non posse confitemur [AT VIII-1, 18-25]. A discussion, relevant to our concerns, of this passage in Wilson (Wilson 1999, p. 111). The translation is hers.
cfr. Plato, [Timaeus, 49d-e]: “(...) that in which they [the properties] each appear to keep coming to be and from which they subsequently perish, that is the only thing to refer to by means of the expressions ‘that’ and ‘this’ ” (Henning 2008).
(Benveniste 1966, vol. 1, pp. 225–236) Position in space and time is a “necessarily individuating property”. Determining such a position involves an essential and ineliminable reference to another individual or position (...) To pick these out as the unique individuals or positions that they are we have to be able to relate them to ourselves or to the here-and-now. (Quinton 1973, pp. 46–50)
An excellent discussion of the perplexities involved in M.D. Wilson, “Superadded Properties; The Limits of Mechanism in Locke”, followed by a “Reply to M. R. Ayers”, in Mechanism, especially p. 212.
This notion is used here in its general sense of the demands that a theory’s truth impose on the world, cfr. (Rayo 2007).
DG in what follows. An edition with translation is available in Rupert Hall and Boas Hall (1962).
The reason for this I discussed in my forthcoming Gravitation-paper. A pre-print is on the ArXives (Verelst 2010).
Burtt (1999, p. 227) One should bear in mind that in the first edition of the Principia these “regulae” were still called “hypotheses”, and had in important respects a different content. See Cohen’s and Chaudhury’s papers on this, discussed below.
Witness Huygens who refers to it as (...) Newton dans ses Principes de Philosophie, que je scay estre dans l’erreur (...) in a letter to Leibniz concerning “true motion” in a discussion involving different notions of causality (Huygens 1888–1950, OH X, n 2854, p. 614 ) (our bold). Huygens’s reasons for this, by the editeurs of the Oeuvres Complètes (in ft 47) rightfully labelled, “assertion remarquable” will be the subject matter of a forthcoming paper.
ULC Add. 3965.14, fl. 613r. This manuscript has been published in facsimile and transcribed with translation by . Ruffner, as Propositiones de Cometis, (2000, pp. 260–263).
The parallellism between space and time had been advanced at first by Gassendi, as pointed out by Bloch (Bloch 1971, p. 179). For a discussion, see De Smet en and Verelst (2001), and McGuire’s discussion of this influence in his “Existence”-paper McGuire (1978a). Newton knew Gassendi’s works since his youth through a book by W. Charleton, and as an adult had several of them in his personal library. Traces of Newton’s reading of Charleton are already found in his Trinity Notebook, dating from his student’s days. It has been edited by McGuire and Tamny (2002, pp. 198–199).
Another set of manuscripts, the Tempus et Locus-texts, relate to this as McGuire points out and elaborates in a comparative study of the different published and unpublished sources concerning this subject, which take up and refine the arguments of the DG McGuire (1978b).
In Janiak’s translation (Janiak 2004, pp. 25–26). I slightly amended the first sentence.
A clear description of the procedure to follow is in Lawden’s book on tensor calculus: (...) the evidence available suggets very strongly that if the motion in a region infinitely remote from all other bodies could be observed, then its motion would always prove to be uniform relative to our reference frame irrespective of the manner in which the motion was initiated. We shall accordingly regard the first law as asserting that, in a region of space remote from all other matter and empty save for a single test particle, a reference frame can be defined relative to which the particle will always have a uniform motion. Such a frame will be referred to as an inertial frame (Lawden 2002, p. 1).
PR, Bk I, Var., p. 54 (16-17). Translation from I.B. Cohen and A. Whitman (Cohen and Whitman 1999, p. 416).
A detailed analysis in H. Poincaré, “La méchanique classique”, in (Poincaré 1902, pp. 112–129).
This difference implicitly codifies the different status rotation and translation have in Newtonian Mechanics. Cfr. Meyerson on this crucial this point: (...) our belief in the homogeneity of space implies something more than the persistence of laws. We are, indeed, convinced that not only laws that is, the relarions between things but even things themselves are not modified by their displacement in space. (Meyerson 1960, p. 37).
Thus, instead of absolute places and motions we use relative ones, which is not inappropriate in ordinary human affairs, although in philosophy abstraction from the senses is required. For it is possible that there is no body truly at rest to which places and motions may be referred. PR, Bk I (Scholium to the Definitions), in the translation by Cohen and Whitman (1999, p. 411).
To the astonishment of several eminent commentators, like Hermann Weyl (1928, p. 71). See on this also Cohen (1966). For this reason I cannot agree with Disalle’s assesment of Leibniz’s criticism (Disalle 2002, p. 40). It is not clear at all what the consequences of even an infinitesimal displacement of the world, given this requirement, would be.
Originally planned to be the second book of the Principia, but which became the third after Newton inserted a new second book in which he gave his final treatment of the original problem dealt with in the De Gravitatione, fluid mechanics. He drafted it in early 1685, but this draft was only published posthumously as Newton’s System of the World (Ruffner 2000, p. 262). For an edition and an explanation of the title, see (Newton 2004, p. xi). More on Book II PR in Smith (2001).
Cfr. Janiak’s interpretation: (...) we ought to distinguish absolute from relative space and time in order to understand true motion as a change of absolute place over time. (...) This move also enables Newton to save the perceptible effects of accelerating bodies—most famously noted in the examples of the rotating bucket and the connected globes in the Scholium—since all accelerations can be understood as true motions within absolute space. (Janiak 2008, p. 50).
This is confirmed rather than refuted by Barbour’s argument on the “Universe at large”, even if he believes it holds for Leibnizian mechanics only: a fully relational (and hence Machain) theory should start by considering the relative motion of the universe treated as a single entity and then recover the motion of subsystems within the background provided by the Universe at large. (Barbour and Bertotti 1982) I believe that Barbour misses a crucial point, however, which prevents him to see that in this respect Newtonian and Leibnizian mechanics ultimately agree. The reason is—again—that he takes only finite, or at best indefinite (countably infinite) “universes” into account. We shall come back to this in the chapter on Leibniz, below.
[CUL add. Ms. 9597.2.11: f.3\(^r\)] See for a discussion and more related material Ducheyne’s paper on the General Scholium (Ducheyne 2006). The quote is in Ducheyne’s translation.
It implies the reverse of the argument on the impossibility to attain by stepwise divisions an actual infinity of parts.
Fatio à W. De Beyrie, pour Leibniz [1694], see Huygen’s Oeuvres Complètes (Huygens 1888–1950, OH X, n\(^{\circ }\)2853, pp. 605–608). This is true beyond question, witness a manuscript in Newton’s hand published by the Halls, the draft of a scholium on corr. 4 and 5 of prop. VI, Book III, where we read: Huius autem generis Hypothesis est unica per quam gravitas explicari potest, eamque geometra ingeniosissimus D.N. Fatio primus excogitavit, (Rupert Hall and Boas Hall 1962, p. 313).
This topic is the central theme of a recent book by Garber (2009), on what he calls the “corporeal metaphysics” of Leibniz’s “middle years”. I came across this book only when this paper was already largely finished, so that its influence remains limited. Evidently, my paper could only have gained from an earlier acquaintance. I nevertheless believe that my findings basically square with the main tenets of Garber’s impressive work.
This text is in the collection provided by Garber & Ariew (Leibniz 1989, pp. 245–250). We use their translation. We do so for all English quotes from Leibniz, except when stated explicitly otherwise.
The reference is to an early work, the Theoria motus abstracti, from 1671. See for this whole period the discussion in Garber (2009).
I.e., composed of impenetrable atoms, as is clear from his reference to the “Epicureans”. The fundamental “elasticity” of matter is an ongoing theme with Leibniz, and a hotly debated issue in the correspondence between Leibniz and Huygens.
Cfr. (Leibniz 1994, p. 42, p. 53). In the Specimen Dynamicum [1691], Leibniz will connect this idea to the existence of the aether, and thus to the cause of gravity: no body is so small that it is without elasticity, and furthermore, each body is permeated by a fluid even subtler than it is. And thus, there are no elements of bodies, nor is there maximally fluid matter, nor are there little solid globes (unintelligible to me) (...) Rather, the analysis proceeds to infinity. (Leibniz 1989, pp. 132–133).
Even though Descartes’s own laws of motion proved wrong (as he himself already suspected), and were corrected afterwards by Huygens, Wren, Wallis and Mariotte, though their solutions remained incomplete. A good overview of the history is in the introduction by Fichant (Leibniz 1994, p. 15 sq.).
This is already in Huygens’s De motu corporum ex percussione [1656]. Whence Leibniz calls it “la méthode du bateau”. Cfr. (Leibniz 1994, p. 14, 31, 190 sq.)
See the Letter to Jean Berthet, Sämtliche Schriften und Briefe [Akademienausgabe], II, vol. 1, p. 383.
Again, this is the origin of his lifelong rejection of atomism, and one of the few fundamental disagreements that continues to surface in his correspondence with Huygens, an important debate to which we shall come back in a another article. Leibniz reiterates this point on several occasions, cfr. e.g. the quote from the Specimen Dynamicum above. (Leibniz 1989, pp. 132–133, p. 136).
The repercussion and bursting apart [of a body after impact] arises from the elasticity it contains, that is, from the motion of the fluid aetherial matter permeating it, and thus it arises from an internal force or a force existing within itself. (Leibniz 1989, p. 135)
An excellent study of Huygens’s approach is Yoder (1988).
Whenever we are dealing with the equivalence of hypotheses, we must take into account everything relevant to the phenomena. (Leibniz 1989, p. 137)
Cfr. Leibniz in the Specimen Dynamicum: I believe that there is no natural truth in things whose explanation [ratio] ought to be sought directly from divine action or will. (Leibniz 1989, p. 125).
Leibniz therefore breaks with another basic tenet of Cartesian mechanical metaphysics: that only the indefinite is real (Belaval 1960, pp. 275–276).
Car Dieu tournant pour ainsi dire de tous côtés et de toutes les façons le système général des phénomènes (...) et regardant toutes les faces du monde de toutes les manières possibles, puisqu’il n’y a point de rapport qui échappe à son omniscience, le résultat de chaque vue de l’univers, comme regardé d’un certain endroit, est une substance qui exprime l’univers conformément à cette vue (...) [Leibniz 1875–1890, II, p. 95]
On the basis of the arguments in this paper, Kochen and Specker will develop their famous theorem for quantum mechanics Kochen and Specker (1967). The original paper, however, deals with the general case of undecidable propositions independently of QM. For the quote: (Specker 1975, pp. 239–246). Translation in C. A. Hooker, see p. 138.
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Verelst, K. Newton versus Leibniz: intransparency versus inconsistency. Synthese 191, 2907–2940 (2014). https://doi.org/10.1007/s11229-014-0465-7
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DOI: https://doi.org/10.1007/s11229-014-0465-7