Laws of nature 9780191063718, 0191063711, 9780191809057, 0191809055

Table of contents :
Institutions and assumptions in the debate over laws of nature / Walter Ott, Lydia Patton --
Early modern roots of the philosophical concept of a law of nature / Helen Hattab --
Laws of nature and the divine order of things / Mary Domski --
Leges Sive Natura / Walter Ott --
Laws and powers in the frame of nature / Stathis Psillos --
Laws and ideal unity / Angela Breitenbach --
Becoming humean / John W. Carroll --
A perspectivalist better best system account of lawhood / Michela Massimi --
Laws: An invariance-based account / James Woodward --
How the explanations of natural laws make some reducible physical properties natural and explanatorily powerful / Marc Lange --
Laws and their exceptions / Stephen Mumford --
Are laws of nature consistent with contingency? / Nancy Cartwright, Pedro Merlussi.

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Laws of Nature

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Laws of Nature

EDITED BY

Walter Ott and Lydia Patton

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors 2018 The moral rights of the authors have been asserted First Edition published in 2018 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2017961506 ISBN 978–0–19–874677–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents Acknowledgments List of Contributors 1. Intuitions and Assumptions in the Debate over Laws of Nature Walter Ott and Lydia Patton 2. Early Modern Roots of the Philosophical Concept of a Law of Nature Helen Hattab 3. Laws of Nature and the Divine Order of Things: Descartes and Newton on Truth in Natural Philosophy Mary Domski

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4. Leges sive natura: Bacon, Spinoza, and a Forgotten Concept of Law Walter Ott

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5. Laws and Powers in the Frame of Nature Stathis Psillos

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6. Laws and Ideal Unity Angela Breitenbach

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7. Becoming Humean John W. Carroll

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8. A Perspectivalist Better Best System Account of Lawhood Michela Massimi

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9. Laws: An Invariance-Based Account James Woodward

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10. How the Explanations of Natural Laws Make Some Reducible Physical Properties Natural and Explanatorily Powerful Marc Lange

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11. Laws and Their Exceptions Stephen Mumford

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12. Are Laws of Nature Consistent with Contingency? Nancy Cartwright and Pedro Merlussi

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Bibliography Index

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Acknowledgments The editors would like to thank Jim Darcy of the University of Virginia for his expert and careful work in assembling the bibliography and standardizing the references. Many thanks to Audra Jenson of Virginia Tech for preparing the index. We are particularly grateful to Peter Momtchiloff of Oxford University Press for his guidance and encouragement. Thanks to Gayathri Manoharan, who managed the project, to the Press’s copy editor, Dawn Preston, to Chris Bessant for an expert round of proofreading, and to the production and design team at Oxford. We would also like to thank the members of Walter’s graduate seminar on laws of nature at the University of Virginia in the spring of 2016: Nazim Adakli, Jim Darcy, Torrance Fung, Lily Greenway, Kirra Hyde, and Mason Pilcher. Some contributors graciously agreed to allow the seminar to read their papers. The graduate students’ thoughts contributed to our comments to those authors. Most of all, we are indebted to the distinguished contributors themselves.

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List of Contributors A N G E L A B R E I T E N B A C H University Lecturer and Fellow of King’s College, University of Cambridge J OHN W. C ARROLL Professor of Philosophy, North Carolina State University N ANCY C ARTWRIGHT Professor of Philosophy, Durham University and University of California, San Diego M ARY D OMSKI Associate Professor of Philosophy, University of New Mexico H ELEN H ATTAB Associate Professor of Philosophy, University of Houston M ARC L ANGE Professor of Philosophy, University of North Carolina, Chapel Hill M ICHELA M ASSIMI Professor of Philosophy of Science, University of Edinburgh P EDRO M ERLUSSI Research Student, Durham University S TEPHEN M UMFORD Professor of Metaphysics at Durham University and Professor II at Norwegian University of Life Sciences W ALTER O TT Professor of Philosophy, University of Virginia L YDIA P ATTON Associate Professor of Philosophy, Virginia Tech S TATHIS P SILLOS Professor of Philosophy of Science and Metaphysics, University of Athens J AMES W OODWARD Distinguished Professor of History and Philosophy of Science, University of Pittsburgh

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1 Intuitions and Assumptions in the Debate over Laws of Nature Walter Ott and Lydia Patton

Few concepts are as malleable as that of a law of nature. Until the seventeenth century, the phrase was typically a rhetorical device for lauding the apparent orderliness of the non-human world.¹ St. Thomas Aquinas, for example, remarks that ‘if we were to enter a well-ordered house, we would gather from the order manifested in the house the notion of a governor.’² The orderly arrangement of objects and their powers is a testament to the wisdom and benevolence of a creator. As Aquinas goes on to say, ‘the very notion of government of things in God, the ruler of the universe, has the nature of a law.’³ In this usage, it doesn’t make sense to speak of individual laws, such as the law of inertia. Nor is there any prospect of investigating nature to discover the laws it obeys. That pre-modern talk of laws should be toothless in this way makes sense, once one considers the orthodox view that forms its backdrop, at least in the West during the medieval and late medieval periods. Very roughly, the dominant position holds that the course of events is determined by the powers that bodies have. For all their disagreements, philosophers such as Aristotle, Aquinas, and Suárez all hold a version of this view. To try to put talk of laws to any explanatory or predictive use would be to have one thought too many: there is already a metaphysical structure in place that underwrites these epistemic practices, and it appeals only to powers.⁴ The notion of a law of nature first gets something like its current sense in the seventeenth century, in the work of René Descartes. In a 1630 letter to Marin Mersenne, Descartes claims that

¹ For the origin of the concept of a law of nature, see esp. J.E. Ruby (1986), John Milton (1998), Friedrich Steinle (2002), and Sophie Roux (2011), as well as the chapters by Helen Hattab and Stathis Psillos in this volume. ² Summa Theologicae (henceforth ‘ST’) I q.103 a.1, in Aquinas (1945). ³ ST I q.91 a.1. ⁴ We are of course exaggerating the degree to which philosophers over this vast span of time belonged to a unified orthodoxy. Still, some of the most commonly cited heterodox thinkers—such as Nicholas of Autrecourt and Ibn Al-Ghazali—strike us as being chiefly concerned with the epistemology of powers rather than the metaphysics.

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[t]he mathematical truths which you call eternal have been laid down by God and depend on him entirely no less than the rest of his creatures. Indeed to say that these truths are independent of God is to talk of him as if he were Jupiter or Saturn and to subject him to the Styx and the Fates. Please do not hesitate to assert and proclaim everywhere that it is God who has laid down these laws in nature just as a king lays down laws in his kingdom.⁵

Although Descartes is here speaking of the truths of mathematics, he extends his claim to physics in the Principles of Philosophy, as we shall see. For now, the crucial point is that Descartes’s God stands outside of the nature his laws are to govern. Unlike the Aristotelians, Descartes does not think that nature determines its own course. It has to be directed from without. This is the chief innovation Descartes’s appropriation of ‘law’ talk is designed to achieve. For many Aristotelians, God plays the role of king ruling over nature, but only in a mediate way. On their view, God functions as the first and primary cause of all events, in that God is the source of all being. But creatures nevertheless have their own powers, which function as secondary causes. As Aquinas puts matters, ‘[t]he whole effect proceeds from [both God and the natural agent], yet in different ways, just as the whole of one and the same effect is ascribed to the instrument, and again the whole is ascribed to the principal agent.’⁶ For Suárez as for Aquinas, God gets to decide what happens only in the sense that he is responsible for creating bodies and concurring with their powers.⁷ For Descartes, by contrast, God directly determines the course of events, and ‘lays down’ a ‘law’ that is not fixed by the nature of the objects that ‘obey’ it. Descartes bends the scholastic framework of primary and secondary causes to his novel ends. In the Principles, he claims that, although God is the universal and primary cause, the laws of nature are ‘the secondary and particular causes of the various motions we see in particular bodies.’⁸ On the scholastic view, secondary causes are needed to diversify the being that God creates. God creates and preserves things, but their precise natures are due to the secondary causes that unfold over time. That a given cat exists at all is due to God’s providing it with being; but that it exists as a cat and not as a mouse or a doorstop is due to the powers and hence natures of the created beings that conspired to give it birth. Cartesian secondary causes play the same role: without them, God’s effects would

⁵ AT I 145/CSM III 23. References to Descartes are to the Cottingham, Stoothoff, and Murdoch translation (CSM) and to Adam and Tannery’s edition of Descartes’s work (AT). ⁶ Summa Contra Gentiles in Aquinas (1945, vol. ii, 130). In the same work, Aquinas explains that ‘The order of effects is according to the order of causes. Now the first of all effects is being, for all others are determinations of being. Therefore being is the proper effect of the first agent, and all other agents produce it by the power of the first agent. Furthermore secondary agents which, as it were, particularize and determine the action of the first agent, produce, as their proper effects, the other perfections which determine being’ (1945, vol. ii, 119). ⁷ For Suárez, see esp. Metaphysical Disputation 19, 1 in Suárez (1994, 281–2). ⁸ AT VIIIA 62/CSM I 240.

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remain undifferentiated.⁹ But what plays this role is not a body with a power of its own; it is a law. Descartes’s innovation opens up the conceptual space needed to investigate particular, determinate laws that can be expressed mathematically. It would be hard to exaggerate what a departure this is from the scholastic model of scientific investigation which, for all its variations and innovations over the centuries, remained largely yoked to the model of explanation by classification in terms of powers and natural kinds. Philosopher-scientists such as Robert Boyle complain that the Cartesian concept of lex naturalis is merely metaphorical: ‘to speak properly, a law being but a notional rule of acting according to the declared will of a superior, it is plain that nothing but an intellectual being can be properly capable of receiving and acting by a law.’¹⁰ And yet even Boyle himself was soon using the phrase. Boyle’s predicament is hardly unusual. From the start, nearly everyone who bothers to reflect much on the concept recognizes it as a legal-cum-theological metaphor that needs to be cashed out. This makes the topic importantly different from other philosophical notions such as responsibility or justice. In those cases, it does seem that pre-theoretical commitments and beliefs shape our subsequent reasoning, and put constraints on the kinds of conclusions we are willing to draw. Such is not the case, we believe, with the idea of a law of nature. That is a highly artificial notion introduced by Descartes to play a very specific role in his philosophy of physics and theology. Later thinkers will of course bend the notion to their own purposes. If the concept of a law of nature is artificial in this way, we must be careful how we proceed. In thinking about moral responsibility, for example, it makes sense to weigh and sift through our intuitions and aim for a view that achieves the maximal degree of reflective equilibrium. One tries to preserve the strongest intuitions and achieve some kind of coherence among them. But when our intuitions and assumptions are historically conditioned in the way we believe nomological commitments are, this procedure is dubious. We must first become aware of the provenance of our intuitions: are they genuine insights, or the relics of a theological world view? Are they an artifact of (possibly outmoded) scientific practices, or simply the result of focusing on a narrow set of examples of such practices? This chapter aims to sift through the intuitions that have guided the debate over laws of nature. It would be a fallacy, of course, to assume that an intuition’s origin in an outmoded theory or world view is a mark against it. Rather, our point is that only by conducting such a genealogy can we see that these intuitions are not permanent and necessary features of everyone’s conceptions of the world. They do not automatically deserve a place in our theorizing about laws; each must live or die on its own merits. We begin with the guiding intuition of Descartes’s founding account, that laws govern events. ⁹ For a different reading of Descartes on secondary causes, see Helen Hattab (2000) and (2007). ¹⁰ ‘A Free Inquiry into the Vulgarly Received Notion of Nature,’ in Boyle (1991, 181).

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1. Laws Govern That laws in some sense ‘govern’ the events they are about is a feature of the earliest modern use of the phrase. For Descartes, fixing the particular facts that make up what David Lewis calls the ‘Humean mosaic’ does not fix the laws of nature. Even more broadly: nothing about the world of extension, not the essences of things in it or the powers of bodies, can be used to derive the laws of nature. This is because those laws have their source, not in the created world, but in God: as Descartes puts it in the Principles of Philosophy, ‘[f]rom God’s immutability we can . . . know certain laws or rules of nature, which are the secondary and particular causes of the various motions we see in particular bodies.’¹¹ We can call this the ‘top-down’ view: laws are imposed, as it were, from above. The top-down view has immediate consequences for causation: there is no way for bodies to have genuine causal powers.¹² In 1678, Ralph Cudworth makes the argument explicit when he attacks ‘those mechanic Theists’ who affect to concern the Deity as little as is possible in mundane affairs, either for fear of debasing him, and bringing him down to too mean offices, or else of subjecting him to solicitous encumberment; and for that cause would have God to contribute nothing more to the mundane system and economy, than only the first impressing of a certain quantity of motion upon the matter, and the after conserving it, according to some general laws; these men, I say, seem not very well to understand themselves in this. Forasmuch as they must of necessity, either suppose these their laws of motion execute themselves, or else be forced perpetually to concern the Deity in the immediate motion of every atom of matter throughout the universe, in order to the execution and observation of them. The former of which being a thing plainly absurd and ridiculous, and the latter, that which these philosophers themselves are extremely abhorrent from, we cannot make any other conclusion than this, that they do but unskilfully and unawares establish that very thing, which in words they oppose [i.e., the hypothesis of a plastic nature].¹³

The problem Cudworth isolates is this: there is no way for God to decree or set down a law of nature without providing the means for its enforcement. Laws cannot ‘execute themselves.’ Anyone who wants to retain the top-down view, Cudworth argues, faces a choice: either ratchet up the capacities of mere matter, so it can in fact understand and obey God’s laws, or involve God in every causal transaction. Although it seems not just surprising but comical from our perspective, Cudworth chooses the first option. His ‘plastick nature’ is a sort of ‘deputy God’¹⁴ who puts the laws into action. ‘Nature,’ Cudworth writes, ‘is art as it were incorporated and embodied in matter, which doth not act upon it from without mechanically, but from within vitally and magically.’¹⁵

¹¹ AT VIIIA 62/CSM I 240. ¹² This argument is made at greater length in Ott (2009), ch. 9. ¹³ Cudworth (1837, vol. 1, 213–14). ¹⁴ The phrase is Jesseph’s (2005). ¹⁵ Cudworth (1837, vol. 1, 220).

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Such a view is of course anathema to Cartesian mechanists, who attribute only size, shape, and motion to bodies. For Descartes himself, it seems fairly clear that his conception of laws and bodies leads to at least a kind of limited occasionalism. Although finite minds might be causes, where bodies are concerned, God is the only cause.¹⁶ How else could the laws of nature follow from God’s nature and will? As Cudworth argues, it is not as if the laws, once decreed, attain a kind of independent existence and can march about the universe directing the traffic of bodies. Hence topdown visions of laws in the modern period are tied to occasionalism. Although top-down views have their origins in theism, they of course appear in the twentieth century in a secular context. Whether they can survive the transplant remains to be seen. Let us consider briefly the most prominent top-down view in the last forty years, which was independently devised by Fred Dretske, Michael Tooley, and David Armstrong. On the ‘DTA’ view, laws are relations among universals. To say that it is a law that Fs are followed by Gs is to say that the instantiation of F nomically necessitates the instantiation of G, or raises the probability of such an instantiation.¹⁷ As Armstrong puts it, the necessitation relation is ‘like an inference in nature.’¹⁸ This view rejects the claim that the laws are fixed by the aggregate of local matters of fact and yet tries to avoid both theism and animism. A key question here is just what DTA’s nomic necessitation amounts to. It is not logical necessitation, since defenders of DTA want to maintain the contingency of the laws. In some worlds, N(F,G) holds, and in others, it doesn’t. It might not hold even in a world where Fs are always coupled with Gs. Armstrong claims that we have to admit this sui generis notion ‘in the spirit of natural piety.’¹⁹ For DTA’s Humean opponents, the well of such piety runs dry at this point. For one might wonder whether there isn’t something purely stipulative about assigning the necessitation relation to universals one finds constantly conjoined. Those attracted to top-down views have another option: non-reductivism. Perhaps the whole project of cashing out the legal-cum-theological metaphor is ill-conceived.²⁰ Isaac Newton famously refuses to speculate (in print, at least) on the ultimate source or underpinning of laws of nature.²¹ Perhaps there is no such source; to try to understand laws as aspects of God’s will or as necessitation relations among universals is to try to analyze a primitive. Although not without its attractions, this view seems to require one to adopt a strange ontological stance. We are at once asked

¹⁶ This claim is highly controversial; for a defense, see Garber (1993) and Ott (2009). For a different take on Descartes’s concept of law, see Helen Hattab’s chapter in this volume. Other relevant literature is cited in Hattab’s chapter. ¹⁷ See Armstrong (1983, 88). ¹⁸ Armstrong (1997, 232). ¹⁹ Armstrong (1983, 92). For further criticism along these lines, see esp. Barry Loewer (2004, 196 f.). ²⁰ John Carroll (1994) defends a non-reductive view, as does James Woodward, in his own way (this volume). Section 3 of Carroll’s chapter for this volume explores the question of whether laws govern. ²¹ See Newton (2004, 63 f.). For more on Newton’s method, see esp. Smith (2002) and Stein (1990), as well as the chapters by Mary Domski and Stathis Psillos in this volume.

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to admit laws into our ontology and prohibited from asking just where they can fit. If they are not relations among universals, what are they? And how can they make it necessary, in whatever sense, that nature take the course it does? We emphasize the question how to understand nomic necessitation because it gets to the heart of the top-down view. All top-down views have it that the laws are actually doing something. If laws are divine volitions, then it is easy to see how this works: God, being omnipotent, forms volitions that are necessarily effective. It is much less easy to see how laws could govern anything on non-theistic top-down views: what does it mean to say, for example, that one universal necessitates the instantiation of another? It is still harder to conceive how laws could be ground-floor elements of our ontology, floating free of the bodies that ‘obey’ them. In the end, it is hard to escape the suspicion that the governing intuition is a holdover from the age of theism.²² If we resist that intuition, one of the chief motivations for the top-down view falls away.

2. Laws Explain That laws govern is only one intuitive source of support for the top-down view. Another such intuition, for Descartes as much as for DTA, is the perhaps more deeply felt intuition that laws explain their instances. In coming to know a law, one comes to know more than the parts of the mosaic the law governs: one knows why the mosaic is as it is. Without governing laws, there is no obvious way in which the laws can explain their instances.²³ Here again, it seems to us that what we have is not a timeless insight but a historically conditioned and indeed shifting criterion. Part of what explains that shift is a change in epistemic ambition and a desire to purge natural philosophy of its dependence on theology. The key figures in the modern debate over explanation are George Berkeley and David Hume. Although Hume is less explicit than one would like, his treatment of causation provides a convenient starting point. On one of Hume’s definitions of the term, a cause is ‘[a]n object, precedent and contiguous to another, and where all the objects resembling the former are plac’d in like relations of precedency and contiguity to those objects, that resemble the latter.’²⁴ This suggests that a law of nature will ²² This is the view defended in Ott (2009). Others are bound to have had the same general suspicion. ²³ This is a point on which Armstrong lays considerable weight; see his (1983, chs. 2–5) and Loewer’s response (2004). ²⁴ A Treatise of Human Nature Book I, Part I, section 14, in Hume 1739–40/2000, 114. Correspondingly, Hume defines necessity ‘in two ways, conformable to the two definitions of cause, of which it makes an essential part. I place it either in the constant union and conjunction of like objects, or in the inference of the mind from the one to the other’ (Book II, Part III, section 2, in Hume 1739–40/2000, 263). It is important not to be misled by Hume’s gloss on the first definition of cause in the Enquiry. There, he says that a cause is such that ‘if the first object had not been, the second never had existed’ (section 7 in Hume 1748/2006, 146). David Lewis (1973, 556–7) and Peter Menzies (2014) construe this as a counterfactual analysis of causation. This would be disastrous, since counterfactual dependence is neither equivalent to,

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simply be a statement of a regularity. The law does not govern anything; it merely summarizes what happens. A crude identification of laws with regularities quickly runs into trouble. For example, some laws are stated in terms of ideal conditions and do not describe any regularities at all. Arguably, the law of gravity doesn’t summarize regularities simply because it ignores the operation of other forces that are always present to some degree.²⁵ Conversely, there are lots of regularities—night following day, for instance—that no one would want to count among the laws. Finally, it seems obtuse to explain a given event by simply pointing to all of the other events that are similar. Regularities do not explain; they are the things to be explained. For a more sophisticated view, we need to back up slightly and consider George Berkeley’s position in De Motu.²⁶ Far more than Hume, Berkeley was sensitive to the actual practices of the scientists of his day. For Berkeley, science is not engaged in tracking causes at all, nor are laws statements of causal relations. As Berkeley puts it, it is not ‘in fact the business of physics or mechanics to establish efficient causes, but only the rules of impulsions or attractions, and, in a word, the laws of motions, and from the established laws to assign the solution, not the efficient cause, of particular phenomena.’²⁷ These rules need not be regularities. Newton’s laws, for instance, need not summarize individual events that actually happen. Instead, they can be principles or theorems that, whether alone or in combination, allow one to deduce the course of events. To explain an event is not to place it in a series of regularities. Instead, ‘[a] thing can be said to be explained mechanically then indeed when it is reduced to those most simple and universal principles, and shown by accurate reasoning to be in agreement and connection with them.’²⁸ Hence David Lewis’s contemporary ‘Humeanism’ would be better termed ‘Berkeleyanism.’ On Lewis’s view, as on Hume’s and Berkeley’s, there is no genuine mind-independent necessity knitting together events, or the universals that figure in them (as DTA would have it). But it is Berkeley, not Hume, who identifies laws as general statements that play a role in scientific practice. And this is the key move in Lewis’s theory. If we suppose that there is, or will be, a single deductive system that best describes the world, then Lewis can define a law as a ‘regularity [that is] a theorem of the best system.’²⁹ As we’ve just seen, the Humean is probably better off not making laws a subset of regularities at all but simply treating them as the theorems of the best

nor does it follow from, constant conjunction. Anne Jaap Jacobson (1986) has shown that this is not Hume’s meaning; in Hume’s English, his claim really is a restatement of the constant conjunction definition and not a counterfactual claim at all. ²⁵ ²⁶ ²⁷ ²⁸

Nancy Cartwright famously makes this point in (Cartwright 1980). See Psillos (2002) for a sophisticated contemporary version of the regularity theory. De Motu section 35 in Berkeley (1721/1975, 218). De Motu section 37 in Berkeley (1721/1975, 218). ²⁹ See Lewis (1994, 478).

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system. Barry Loewer, for example, points out that Lewis’s system can accommodate vacuous laws, which are regularities only in a Pickwickian sense.³⁰ Lewis goes well beyond Berkeley in exploiting the metaphysics of possible worlds. And this allows him to recapture some aspects of the intuition that laws are sources of explanation. Consider the closely related issue of counterfactual support: if the law that Fs are Gs supports the counterfactual ‘had x been F, it would also have been G,’ there is a legitimate sense in which its being a law that Fs are Gs explains the actual distribution of Fs and Gs in the mosaic. Defenders of the top-down view will object that mere regularities do not support counterfactuals: the fact that everyone in this room is wearing shoes hardly suggests that if a barefoot person were here, shoes would magically appear on her feet.³¹ But if laws are not regularities (or statements of them) but instead theorems of the best system, then we can reclaim the support of counterfactuals. Any nearby world will by definition be one in which the laws hold, and hence any nearby world in which Fa is one in which Ga as well. Assuming a truth-functional account of conditionals, this lets us say that had a been F, it would also have been G. Now, the anti-Humean is still free to complain that this is not genuine support of the counterfactual. There is surely something bizarre, perhaps question begging, about assuming that the closest possible worlds share our laws of nature on one hand and collapsing those laws into sophisticated summaries of events on the other. Many attacks on Lewis-style views have exactly that structure, arguing that Lewisian ‘explanation’ and ‘counterfactual support’ are mere counterfeits and not the genuine articles.³² It is worth wondering, though, whether the defender of the best systems analysis needs to respond by recasting these concepts in a Lewisian mold. Why not simply say that they are relics of the top-down picture? Absent an argument to the effect that only the top-down, governing conception provides explanation and counterfactual support worth wanting, the Berkeleyan view’s withers are unwrung.

3. Laws Explain, Round 2 The claim that laws explain in a robust sense has so far been exhibited as a star in the top-down constellation of thought. Many philosophers have felt its gravitational pull, however, without being at all attracted to the top-down view. For these thinkers, to treat the Humean mosaic as a stopping point is unacceptable. For we can sensibly ask, what explains the Humean mosaic itself? We can treat laws as mere summaries but then ask, what explains the events being summarized? Consider the consequences of the Lewisian view for our understanding of objects and their properties. Such a position has to deny bodies and their properties any ³⁰ See Loewer (2004). ³¹ The locus classicus for this kind of argument is Armstrong (1983, 46–51). ³² See Loewer (2004, 189 f.) for an illuminating discussion of this dialectic.

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genuine causal role. For Hume, this point is straightforward: there is nothing more to causation than constant conjunction, except perhaps the felt need to make the transition from one perception to another. For Berkeley, only God and perhaps created minds are causes. Bodies are ideas which are by their nature causally inert. For Lewis, as for Hume and Berkeley, all properties are quiddities, that is, properties that are categorical only, with no power to change the course of events. This characterization is of course controversial, for Lewis develops his own counterfactual account of causation, which would allow properties to count as causes in that sense. But what matters here is that Lewis, just as much as Hume, denies that there are any mind-independent powers in the ordinary sense of that term. Sydney Shoemaker famously argues that if there were quiddities, we would have no way of knowing about them. For our perceptual apparatus only tracks causal powers, and quiddities by definition escape our perceptual abilities.³³ One need not find that line of argument decisive to feel the pull of the demand for non-Berkeleyan explanation. To accommodate it, philosophers hostile to the top-down constellation are trying to turn the clock back to the time of the ancient Greeks. On this view, what makes nature take the course it does is the powers or dispositions had by the objects in it. Although the view is increasingly common in the twenty-first century, its contemporary revival can be seen in Rom Harré and E.H. Madden (1975).³⁴ What notion of a ‘law of nature’ best fits this view? The field of candidates is broader than one might have thought. The powers view is resolutely bottom-up, and so Armstrong-style analyses, as well as anti-reductionism, are off limits. But nothing stops the proponent of powers from embracing the Berkeleyan story, or its contemporary Lewisian variant. Such a theory would hold that laws are theorems of the best axiomatization of the Humean mosaic. It’s just that the powers possessed by the objects that figure in the mosaic are ultimately responsible for the mosaic itself. One would of course have to reject the counterfactual analysis of causation and other Lewisian accretions, but the core of the Berkeleyan analysis would remain intact. There is an alternative emerging in the current literature, one with an even more ancient pedigree. In the work of Spinoza and Bacon, one finds a competing use of the term ‘law.’ On their views, talking about laws is a way of talking about dispositions.³⁵ In the contemporary scene, Brian Ellis speaks of ‘a law of action . . . that describes what [a given power] does when it acts.’³⁶ This notion is miles away from the Cartesian theological metaphor. It might accommodate the intuition that laws explain in a robust sense while staying well clear of the top-down family of views. ³³ This is a very rough version of the argument; see Shoemaker (1980). ³⁴ Among the other figures in this Aristotelian renaissance are Brian Ellis, Ruth Groff, and Stephen Mumford. ³⁵ See Ott’s chapter in this volume; Stephen Mumford’s chapter also uses ‘law’ in something like this sense. ³⁶ Ellis (2010, 136).

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4. Laws Enable Prediction The logical empiricist tradition is the source of much theorizing about laws in the philosophy of science, of characteristic definitions of ‘explanation’ and ‘law,’ and of a profoundly influential theory of the relationship between science, logic, and philosophy. Any project of sifting through intuitions about laws must sieve the logical empiricist tradition in turn. Wesley Salmon has argued that Rudolf Carnap, Moritz Schlick, Hans Reichenbach, and their fellow logical empiricists divorce scientific explanation from the pragmatic aims of prediction and control. Mary Hesse refers to the ‘pragmatist criterion’ for progress in science, that scientific theories and experimental methods allow for increasingly successful prediction and control.³⁷ On Salmon’s reading, the ‘Received View’ associated with the logical empiricists emphasizes the pragmatist criterion to the exclusion of scientific explanation, which is treated as metaphysical. In 1988, Salmon writes, During the last forty years, few (if any) have voiced the opinion that the sole aims of science are to describe, predict, and control nature—that explanation falls into the domains of metaphysics or theology. It has not always been so. Twentieth century scientific philosophy arose in a philosophical context dominated by post-Hegelian and post-Kantian German Idealism. It was heavily infused with transcendental metaphysics and theology. The early logical positivists and logical empiricists saw it as part of their mission to overcome such influences.³⁸

Salmon’s criticisms of the ‘received view’ among logical empiricists reflect a broad consensus. The received view is widely accepted to be influential but untenable, although recent work aims to rehabilitate the program.³⁹

³⁷ See Hesse (1980). The terms ‘prediction’ and ‘control’ have a long history in behaviorist psychology, beginning with John Watson and B.F. Skinner. Richardson (2006) analyzes the terms as they appear in Reichenbach’s Experience and Prediction (1938), and Cartwright et al. (2008) contains a number of discussions of the role of prediction and control in Neurath’s thought. These sources urge that prediction and control have epistemic import (in Reichenbach’s case) or otherwise are guides to action (in Neurath’s case). Richardson (2006): ‘science will achieve objectivity . . . through a demand for epistemic control; science seeks claims that can be checked against the world and which epistemic agents can agree upon (so we can check one another)’ (2006, 46). See Neurath’s remark: ‘Carnap . . . distinguished two languages: a “monologizing” one (phenomenalist) and an “intersubjective” (physicalist) one . . . only one language comes into question from the start, and that is the physicalist. One can learn the physicalist language from earliest childhood. If someone makes predictions and wants to check them himself, he must count on changes in the system of his senses, he must use clocks and rulers, in short, the person supposedly in isolation already makes use of the ‘intersensual’ and ‘intersubjective’ language. The forecaster of yesterday and the controller of today are, so to speak, two persons’ (Neurath 1931/1983, 54–5, cited in Cat 2014). ³⁸ Salmon (1989, 4). Salmon’s statement that the logical empiricists were aiming to overcome ‘transcendental metaphysics’ is only partly true. On the neo-Kantian contexts of logical empiricism, see Richardson (1998) and Friedman (1999). One might also challenge Salmon’s implicit assertion that, to the logical empiricists, description, prediction, and control are distinct from scientific explanation. ³⁹ See, e.g., Lutz (2012). Recent work by Psillos cites the theories of causality and explanation found in Herbert Feigl, one architect of the received view.

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Salmon argues that the logical empiricists attempted to answer only ‘what’ questions, not ‘why’ questions. He puts Rudolf Carnap’s project of conceptual explication under this rubric: on Salmon’s reading, Carnapian explication replaces a vague and unclear concept with a clear and simple one, but does not answer any deeper questions about the meaning of the concept.⁴⁰ He has similar objections to the influential view of laws in logical empiricism articulated by Carl Hempel and Paul Oppenheim: the covering law model of explanation and the associated Deductive-Nomological (D-N) model. Many classical problems (Goodman’s new riddle of induction, for instance) are problems that arose for the covering law model.⁴¹ On Salmon’s reading, Hempel’s D-N model reveals only the ‘logical relation between premises and conclusion’ that ‘shows that the former explain why the latter obtained.’⁴² Once we have established the truth of the conclusion, the D-N model gives only a kind of regressive justification, which demonstrates how the conclusion follows from the laws and inferences within the model. To Salmon, the D-N model is of a piece with Carnapian explication in that it does not explain why the conclusion is true. Rather, it reveals the logical structure of the inferences used to reach that conclusion, and thus explains the reasons why the conclusion is said to be true— which is different from explaining why the conclusion is true. Assuming this reading to be correct, is it true that the logical empiricists cannot account for the intuition that laws explain? Certainly, according to Salmon’s own criteria, logical empiricist laws fail to explain why things happen as they do. For instance, the laws in the D-N model have a certain necessary form, but that form was only to account for the role played by the laws in inferences.⁴³ Thus, it is claimed, the ‘Received View’ does not capture our intuition that laws explain their instances. Which ‘laws,’ though? One often overlooked aspect of the Received View is that it does not identify scientific laws with Cartesian laws of nature. Metaphysics generally must be divorced from our account of laws. Among the key influences on logical empiricism were the views of Ernst Mach (one wing of the Vienna Circle was the

⁴⁰ Salmon (1989, 5–6). ⁴¹ Salmon (1989) provides an overview of these problems, of the model itself, and of the ‘new consensus’ on the notion of explanation in science. The ‘old consensus’ was reached in the heyday of logical empiricism, on the basis of Hempel and Oppenheim (1948). For critical remarks on the history presented in Salmon (1989), see below. The notion of theories as representations has been influential as well in the structuralist and semantic view of theories; see Da Costa and French (2000) for an overview. ⁴² Salmon (1989, 7). ⁴³ On a standard reading of the ‘Received View,’ scientific theories must be fully axiomatized, with all results in the theory following strictly from axioms and inference rules. Lutz (2012, 77): ‘In this view, a scientific theory is formalized as a set of sentences (called theoretical sentences) of predicate logic that contain only logical or mathematical terms and the terms of the theory (theoretical terms). The theoretical terms are connected to terms that refer to observable properties (observation terms) through sets of correspondence rules, sentences that contain both theoretical and observation terms. The observation terms are given a semantic interpretation, which, through the correspondence rules and theoretical sentences, restricts the possible semantic interpretations of the theoretical terms’ (Carnap 1939, sec. 24; Feigl et al. 1970, 5–6). Lutz mounts a qualified defense of the Received View.

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‘Ernst Mach Society’) and the conventionalist accounts of Henri Poincaré and Pierre Duhem. Mach argued that the laws of nature contribute to the ‘economy’ of science, a notion that is receiving increased attention. For Mach, the laws may promote aims we have for science—control, transparency, calculating power—but do not provide an explanatory framework that goes beyond observable phenomena. Duhem raises a difficulty with the ‘Cartesian method’ in science, precisely because it combines metaphysics with physics: The physicist who wishes to follow [the Cartesians and atomists] can no longer use the methods proper to physics exclusively . . . Here he enters the domain of cosmology. He no longer has the right to shut his ears to what metaphysics wishes to tell him about the real nature of matter; hence, as a consequence, through dependence on metaphysical cosmology, his physics suffers from all the uncertainties and vicissitudes of that doctrine. Theories constructed by the method of the Cartesians and atomists are also condemned to infinite multiplication and to perpetual reformulation. They do not appear to be in any state to assure consensus and continual progress to science.⁴⁴

If scientists must work out the essence of matter and of the universe before they can begin constructing a theory of physics, physics won’t make much progress. The practice of physical science requires that we divorce mechanics from ‘dependence on metaphysical cosmology.’ Neither Duhem nor Mach identifies scientific laws—the laws formulated in mechanics and physics—with Cartesian or Berkeleyan ‘laws of nature,’ and nor do most of the logical empiricists. The notion that philosophy and science should be continuous pilots some logical empiricist ships, but certainly not the notion that metaphysics and science should be. The question whether laws on the Received View explain their instances cannot be answered, then, without also answering the question of whether the laws in question are laws of nature, or scientific laws. If the latter, then there is a question whether the laws employed by the Received View even were intended to be fit for very many of the roles Descartes, Berkeley, and the others wanted laws to play. Since an explicit aim of many logical empiricist philosophers was to separate metaphysics from science, it may be wise to take them at their word, and to concede that the logical empiricist position was never intended to capture the Cartesian or Berkeleyan intuitions about the laws of nature. The more interesting question, then, is: What is the significance for the philosophy of science of the turn away from metaphysical laws of nature and toward scientific laws? On Salmon’s account, the logical empiricists cannot use laws to explain their instances. But Hempel and Oppenheim focus the D-N model precisely on explanation. What seems to be at stake, then, are criteria of adequacy for ‘explanation.’ Apparently, behind Salmon’s objections is the conception that laws should explain ⁴⁴ Duhem (1996/1917, 233–4).

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why events take place, not just by revealing the structure and source of law-governed inferences, but by explaining why the laws are the source of true conclusions about events. If so, Salmon is requiring that an account of laws must satisfy what some logical empiricists would take to be a metaphysical notion of the truth of those laws. We may well ask, then, whether Salmon and the Received View are operating with the same notions of truth or of explanation in the first place. Salmon is correct that banishing metaphysics and ‘theology’ from philosophical reasoning about the laws of nature was a motivation for some logical empiricists. Salmon does not account for the influence of conventionalism, neo-Kantianism, and pragmatism on logical empiricism, an influence that has been tracked in detail by subsequent scholars in the history of philosophy of science. We might trace the relative neglect of these sources to the continuing influence of Willard van Orman Quine’s ‘Two Dogmas of Empiricism,’ starring Carnap as the lead dogmatic empiricist. Quine urged that abandoning logical empiricism would result in a turn toward pragmatism.⁴⁵ The implicit assertion that logical empiricism and pragmatism are at odds with each other has been challenged in recent work.⁴⁶ What changes would be wrought to our account of the Received View if we were to interpret their account of law-governed explanation, and of explication, as motivated by the separation of Cartesian metaphysics from physics, and by the view that laws need not be true to explain?

5. Laws Are Universal There is an enduring intuition that laws of nature must be universal. To paraphrase Sellars, since laws explain nature, they should explain why things, in the broadest possible sense, happen the way they do, in the broadest possible sense. Pragmatist and conventionalist currents in the philosophy of science and in metaphysics challenge this intuition—and some of these currents are found within the logical empiricist tradition itself. Nancy Cartwright’s Why the Laws of Physics Lie, and her work since, has established an enduring challenge to the intuition that laws are universal statements that hold without exception.⁴⁷ It may come as a surprise, then, to learn that Cartwright locates a source of her own view within the logical empiricist tradition. One way to read Cartwright’s ‘dappled world’ is that, in it, prediction and control are local phenomena. Cartwright cites her ‘own hero, Otto Neurath’: ‘Those who

⁴⁵ Quine (1951), 20; see Richardson 2007 for a critical response. ⁴⁶ Richardson (2007) and Misak (2016) have analyzed critically this interpretation of the history of logical empiricism and of the philosophy of science more generally. Creath (e.g., 1995) has challenged the thesis that Quine and Carnap are as far apart as some philosophers allege. ⁴⁷ In their chapter in this volume, Cartwright and Merlussi argue that all of the views on laws currently on offer are consistent with the rejection of universality.

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stay exclusively with the present will very soon only be able to understand the past.’⁴⁸ Cartwright continues: Just as the science of mechanics provides the builder of machines with information about machines that have never been constructed, so too the social sciences can provide the social engineer with information about economic orders that have never been realized. The idea is that we must learn about the basic capacities of the components; then we can arrange them to elicit the regularities that we want to see.⁴⁹

By learning about the capacities of components of systems, we may be able to control, locally, the outcomes of the mechanisms we employ or observe. But there is no ultimate system, no universal set of laws that govern the workings of such systems. With Cartwright and Neurath, we move away, not only from the intuition that laws explain, but also from the view that laws are universal and not local. The notion that explanation is local has found expression in the powers view discussed above, but also in the new concern, in philosophy of biology, with explanation via mechanisms.⁵⁰ Frank Ramsey, who proposed a canonical version of the Best System Analysis (BSA), and Hilary Putnam, who proposed the no miracles argument for scientific realism, created now standard positions on the laws of nature that were influenced by pragmatism. Putnam’s and Ramsey’s accounts are what Cohen and Callender call ‘non-governing’ accounts of laws, according to which ‘there are genuine laws of nature, but . . . they do not govern or produce the events of the world. The mosaic of events displays certain patterns, and it is in the features of some of these patterns that we find laws.’⁵¹ On this view, the laws are embedded in the mosaic—as Earman and Roberts (2005) put it, the laws supervene on the ‘humble facts.’ If the system in question is embedded in the Humean mosaic, there might be various ways to formulate the intuition that the laws of the system are universal. For the laws to be universal, they would need to hold independently of ceteris paribus clauses. Such clauses formulate cases in which the background conditions might vary (if you strike a match, it will light, unless the match is wet). Or, ceteris paribus clauses may specify properties of objects that make the laws fail to hold (if you strike a match, it will light, unless the match’s head is made of clay).⁵² ⁴⁸ Trans. by Cartwright from the citation in Nemeth (1981, 51). ⁴⁹ Cartwright (1999, 124). ⁵⁰ See, e.g., Machamer et al. (2000) and Bechtel and Abrahamsen (2005). A growing body of work challenges the universality of laws. Leuridan (2010) gives an illuminating analysis of the philosophical import of recent work on mechanisms and on non-universal laws. A special issue of Philosophy of Science from 1997 focused work questioning the universality of laws in biology, including essays by Beatty, Brandon, and Sober. Christie (1994) cites evidence from chemistry, and from chemical practice, to argue against universal laws in chemistry. Beed and Beed (2000) make a similar case for the social sciences. Woodward (1992, 2003) and others argue for mechanisms as part of a larger account of complex systems. Mitchell (2000) argues explicitly for pragmatism concerning the laws of nature, and against the universalist conception. ⁵¹ Cohen and Callender (2009, 1). ⁵² The account of laws according to which they are the invariants of scientific theories is found in Eugene Wigner, James Woodward, and Emmy Noether. For a recent defense, see Woodward’s chapter in

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For Earman and Roberts, the ‘humble facts’ are the non-nomic initial conditions and boundary conditions of a physical problem. The laws, in turn, are used to derive differential equations that determine the evolution of a physical system given the initial and boundary conditions. On their view, the differential equations are not different in kind from the boundary conditions.⁵³ The equations describing the evolution of the system are not different in kind from physical facts about the system. Thus, in one way, the laws are ‘universal’ by default. There are no background or initial conditions that would make the laws fail to hold, because the laws simply formulate the regularities that hold of the mosaic. However, the laws of the BSA are not universal in the sense specified by Hempel, namely, that their terms are universally quantified in all possible contexts. The claim ‘For all observed x in the mosaic, y is true’ does not imply that ‘For all x, y is true.’ Certainly, the BSA is intended to use laws for prediction, but it does not have the consequence that the universal quantification works for all possible objects or phenomena, without taking account of the observed facts. Thus, Lewis’s and Ramsey’s accounts leave room for distinct ways of reading the Best Systems Analysis as a theory of ‘universal’ laws.⁵⁴

6. Conclusion Three central and related points have emerged so far. Together, they should make us question our methods and assumptions before entering into the scrum. First, Salmon’s critique of the logical empiricist tradition represents, as far as we can tell, the most common attitude among philosophers. Our assessment of Salmon’s critique suggests that the logical empiricists are not engaged in the project Salmon takes them to be. He assumes they are trying to explain and justify the sorts of laws that figure in the metaphysical tradition, from Descartes to Berkeley, and then complains when they fail. But that failure might make us wonder whether the logical empiricists made the attempt in the first place. Have they missed the target, or were they aiming elsewhere? What’s distinctive of their approach, we argue, is the attempt to sever theorizing about scientific practice from metaphysical concepts and debates. If the physicist this volume. For more on Emmy Noether’s arguments on conservation laws and symmetry principles, see Brading (2001). Woodward’s account focuses on the independence of laws from certain initial conditions. Woodward argues that our ability to divorce lawlike statements from initial conditions via the identification of relevant symmetries is fundamental to reasoning about the laws of nature. Here, Woodward shares a methodological approach with Earman and Roberts’s (2005) ‘New Characterization of the Humean Base,’ of looking for ways to divorce laws from their initial and boundary conditions (2005, 13–17). ⁵³ Earman and Roberts (2005, 14–15). ⁵⁴ Cohen and Callender (2009) propose a ‘better best systems analysis.’ In this volume, Massimi unites the best systems analysis with the perspectival realism proposed by Ron Giere, to argue for a perspectival best systems analysis. Massimi’s system has the advantage that it proposes resolutions to philosophical problems and paradoxes long associated with the history of science.

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declines to ‘enter the domain of cosmology,’ she will hardly be bothered by any slings and arrows that are flung from that domain. If this is right, then the logical empiricist tradition is due for a re-evaluation. The cleavage between laws as they actually figure in scientific practice and what we might call ‘the laws of the philosophers’ suggests a second distinction: that between the causal story involved in any given interaction and the philosophical account of the laws that are concerned in that interaction. For instance, as we argued above, the powers theorist is perfectly able to endorse any of a wide array of stories about laws, from Best Systems Analyses to regularity theories; only top-down views are off limits. This suggests that the powers theory is not, in and of itself, competing for quite the same territory as the other theories of laws. An account of powers can close off some stories about laws; but that account need not itself choose one of the remaining competitors. To see this in practice, consider that the powers account is typically taken as a competitor to Humean supervenience accounts.⁵⁵ It is quite right, of course, that the powers theorist is hostile to the project of doing away with mind-independent causal connections, or substituting Lewisian ones in their stead. But nothing prevents the powers theorist from hijacking a Best Systems Analysis and yoking it to her own ontology. Indeed, the powers theorist might retain something of the flavor of the logical empiricist view, and insist that scientific laws are a different kettle of fish altogether. These two reflections suggest a third: that a degree of skepticism is warranted when positions on laws of nature are attacked for not being faithful to this or that alleged desideratum. What features or facts one selects as ‘data,’ or which constraints or criteria one sees as essential to a theory of laws, has as much to do with the tradition in which one is operating as anything else. From some points of view, universality is absolutely non-negotiable; from others, it’s a fantasy, one that might well be rooted in an early modern theological approach that we see in Descartes. Equally, from one vantage point, it’s simply obvious that laws govern, and the task is to say how they do so. From another, that thought is conditioned by a theological tradition we would do well to slough off permanently. The conception of a ‘law of nature’ is a human product. It was created to play a role in natural philosophy, in the Cartesian tradition. In light of this, philosophers and scientists must sort out what they mean by a law of nature before evaluating rival theories and approaches. If one’s conception of the laws of nature is yoked to metaphysical notions of truth and explanation, that connection must be made explicit and defended. If, on the other hand, one’s aim is to disentangle laws from truth or from explanation, that must be stated and defended as well. If philosophers do not make such assumptions, intuitions, and methodological commitments clear, then it will be impossible to identify the source of disagreement

⁵⁵ See, e.g., Loewer’s excellent (2004, 200 f.).

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in debates about the laws of nature. Are the conflicts rooted in disagreement about the conclusions reached, or do the background commitments of the combatants block any resolution to the dispute in principle or in practice?⁵⁶ We are far from embracing a nihilistic relativism about laws. We firmly believe there are facts of the matter to be discovered. The trick is to be sure we are arguing about the same thing. And not to allow our intuitions to exercise their influence unexamined.

⁵⁶ Do we reach a standoff reminiscent of Wittgenstein’s On Certainty, in which each gives her reasons, but no reason can end the debate? (Wittgenstein 1972; see Kusch 2016 for a discussion of epistemic relativism and skepticism in this context.)

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2 Early Modern Roots of the Philosophical Concept of a Law of Nature Helen Hattab

Systematic studies of the emergence of the modern sense of a law of nature, e.g., by Edgar Zilsel, Edward Oakley, Jane Ruby, and, more recently, by Sophie Roux and Ian McLean are largely historical and terminological in nature.¹ While they cite different reasons, each study indicates that earlier uses of the term differ significantly from René Descartes’s appeals to three universal laws of nature.² Indeed, Roux shows that the classical modern use of the term ‘law of nature’ is not consolidated until the end of the seventeenth century and first appears as an amalgam of earlier uses.³ Mapping out different pre-modern and modern senses of the term so as to trace its historical evolution is an important project that has yet to be completed. My aim in this chapter is to focus more narrowly on Descartes’s concept of a law of nature. By situating it against the backdrop of prior concepts of a law of nature, I examine what makes it distinctively modern and trace some of its philosophical roots. Well before Isaac Newton developed the laws of classical physics, Descartes formulated three rules of motion, which he called ‘laws of nature.’ These are: 1) ‘each thing, insofar as it is simple and undivided, always remains in the same state, as far as it can, and never changes except as the result of external causes,’ 2) ‘every piece of matter, considered in itself, always tends to continue moving, not in any oblique ¹ Relevant publications include: Zilsel (1942), Oakley (1961), Ruby (1986), Roux (2011), and McLean (2008). ² Whereas Zilsel attributes this shift to sociological factors and Oakley to theological doctrines, Ruby highlights that earlier uses of ‘law’ in the mixed mathematical sciences have a different meaning, and are used as synonyms to ‘rule’ or ‘proportion.’ McLean shows that ‘law’ and related terms do not even have a fixed meaning in renaissance natural philosophy and are tied to specific ancient natural philosophies. He highlights that the prevailing view of sublunary nature as admitting exceptions made it difficult to formulate laws in the modern sense and though one does find ‘theses,’ ‘hypotheses,’ ‘theoremata,’ ‘schemata,’ and ‘dogmata’ set out in forms that resemble scientific laws, e.g., by Bartholomeo Keckermann, their content was not normally considered mind-independent (McLean 2008, 31–2, 37–8). ³ Roux (2011).

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path but only in a straight line,’ and 3) ‘if a body collides with another body that is stronger than itself, it loses none of its motion; but if it collides with a weaker body, it loses a quantity of motion equal to that which it imparts to the other body.’⁴ Descartes asserts that these ‘rules or laws of nature’ are known from God’s immutability. Since God is the cause of all matter, along with its modes of motion and rest, and since his action is constant and immutable, Descartes claims it is moreover ‘most reasonable to think that God likewise always preserves the same amount of quantity in motion.’⁵ Descartes further explicitly invokes God’s immutability as the reason for the content of each law of motion. Despite their divine origins, Descartes’s laws of nature have key features of laws of nature developed later in the modern period. First, they are universal as they apply to all motions in the universe: ‘From the first instance of their creation, he [God] causes some to start moving in one direction and others in another, some faster and others slower . . . and he causes them to continue moving thereafter in accordance with the ordinary laws of nature.’⁶ By contrast, the kinds of rules or laws found in the mixed mathematical sciences since ancient times apply only within specific domains, e.g., Archimedes’ law of the lever applies to machines, the objects of mechanical science and Johannes Kepler’s laws to planetary motions, the objects of astronomy. Second, unlike the rules or theorems long in use by mathematicians and sometimes referred to as laws, Descartes’s laws of nature are not mere abstractions but rather play a causal role in nature. Whereas God, by virtue of creating and preserving all motion, is the universal, primary, and general cause of all the motions in the world, Descartes characterizes the laws of nature as ‘the secondary and particular causes of the various motions we see in particular bodies.’⁷ How exactly Descartes conceives of their causal role in relation to God’s causality has been heavily debated by Descartes scholars, including myself.⁸ But that they play some causal role is confirmed by the following passage: ‘And all particular causes of changes, which happen to bodies, are comprised [continentur] in this third law, or at least those [causes] which are themselves bodies.’⁹ In the sentence that follows, Descartes

⁴ CSM I, 140–2. References to Descartes are to the Cottingham, Stoothoff, and Murdoch translation (CSM) and to Adam and Tannery’s edition of Descartes’s work (AT). ⁵ CSM I, 240. The principle of the conservation of motion is also found in Isaac Beeckman’s journal. According to Klaas Van Berkel, ‘There are obvious differences between Descartes’s principle and Beeckman’s which probably did not yet exist in 1618 when Descartes was just a passive listener. For example, one important difference is that Beeckman, as is clear from the quote, meant by quantity of motion the product of the quantity of matter and speed, and not as Descartes later meant, the product of the size of the body and its speed. Beeckman had a primitive concept of mass which implied that there could be different densities in the matter’ (Van Berkel, 1983, 210, translation mine). ⁶ CSM I, 91. ⁷ CSM I, 240. ⁸ Scholars have attributed everything from occasionalism to concurrentism and divine conservationism to Descartes. Recent literature on this topic includes: Clatterbaugh (1995); Des Chene (1996, 341); Hattab (1998, 2000, 2003, 2007); Della Rocca (1999); Garber (2001, 189–220); Pessin (2003); Gorham (2004); Schmaltz (2008); and Machamer and McGuire (2009). ⁹ AT VIII, 65.

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explicitly sets aside inquiry into any powers that human or angelic minds might have to move bodies.¹⁰ From this results a third feature of Descartes’s laws of nature: they are determinative of the all regularities we observe in nature, without exception. As Descartes puts it: For God has established these laws in such a marvelous way that even if we suppose he creates nothing beyond what I have mentioned, and sets up no order or proportion within it but composes from it a chaos as confused and muddled as any of the poets could describe, the laws of nature are sufficient to disentangle themselves and arrange themselves in such good order that they will have the form of a quite perfect world.¹¹

To understand the roots of this early modern ancestor to our concept of a law of nature, I examine the conceptual problem space from which Descartes’s distinctive characterization of his laws of motion as laws of nature grew. As indicated, the laws, rules, and theorems invoked in explanations extant in the mixed mathematical sciences of astronomy, optics, and mechanics do not possess all three features that brand Descartes’s concept of a law of nature as modern. 1) They are not universal; 2) since they belong to mathematics, they were commonly thought to be abstract not physical/causal, and 3) as a result, they were normally taken to describe and help predict regular occurrences in nature without being determinative of them. One finds earlier laws of motion that possess some of the features of Descartes’s laws, most notably in Galileo Galilei’s and Johannes Kepler’s works (though consistent with the mathematical tradition, they refer to them as ‘propositions’ or ‘theorems’ not ‘laws’). Nonetheless, Descartes’s laws are distinctive in possessing all three features. Whereas Descartes’s three laws are universal but not mathematical in nature,¹² both Galileo and Kepler formulate rules or laws of motion that are fundamentally mathematical in nature and narrower in scope.¹³ The principles of Isaac Beeckman’s physical ¹⁰ The translation by Cottingham et al. reads ‘All the particular causes of the changes which bodies undergo are covered by this third law—or at least, the law covers all changes which are themselves corporeal’ (CSM I, 242). Note that ‘continentur’ is translated more weakly than it should be and the following clause is then completely mistranslated. The Latin reads ‘saltem eae quae ipsae corporeae sunt.’ Cottingham et al. take ‘eae’ to refer to changes (mutationes) and then add ‘the law covers all changes’ which is not in the text. But the next sentence in which Descartes qualifies that he will not inquire into the possibility that minds could move bodies makes it clear that ‘eae’ refers back to causes which themselves (ipsae) are bodies (corporae sunt). ¹¹ CSM I, 91. ¹² The specific rules of collision that follow from the third law can be expressed mathematically, however, as Steinle notes, Descartes never calls them laws nor does he call the sine-relation in optical refraction a law (Steinle 2008, 221–2). ¹³ Garber goes so far as to argue that ‘While mathematics enters into Descartes’s natural philosophy from time to time, his treatment of the laws of motion seems to be quite independent of any attempts that he may have made to understand nature mathematically. With Galileo, on the other hand, the application of mathematics to nature is quite central to his project. However, it is not clear that the laws of nature play any substantive role in his account of the physical world.’ He also notes, ‘Galileo has the concept of an overarching law of nature, something that governs reality as a whole. But his mathematical account of the motion of bodies is not conceived in those terms’ (Garber, Forthcoming). Similarly, whereas Grasshof

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explanations, which he also referred to as ‘hypotheses,’ most resemble Descartes’s laws in scope and content. In particular, Beeckman probably had his version of the principle of inertia before Descartes.¹⁴ By 1618, Beeckman also had an atomist theory of matter, which like Plato’s attributed shapes to atoms to explain their characteristics, but without any of the accompanying metaphysics.¹⁵ However, the principles of Beeckman’s physics stem from pragmatic and epistemological considerations as he eschews metaphysical speculation.¹⁶ Descartes’s laws, by contrast, are grounded in his metaphysics, a subject on which, except for Kepler, contemporaries engaged in mathematical explanations were largely silent. I thus turn to the metaphysical roots of Descartes’s more philosophical concept of a law of nature. First, given that Descartes was educated by the Jesuits, who based their natural philosophy primarily on St Thomas Aquinas’ doctrines, I outline the Thomist view of the order and law of nature.¹⁷ This gives us a handle on the traditional picture against which Descartes developed his new natural philosophy. Next I examine the related concepts of fate, universal nature, and its laws discussed by Domingo de Soto, Francesco Piccolomini, and Jacopo Zabarella, three much cited Aristotelian natural philosophers of the late sixteenth and early seventeenth centuries. Finally, I focus on the combination of Stoic and Neoplatonic metaphysical principles in the atomist natural philosophy developed by the early seventeenth-century Calvinist philosopher, Sebastian Basso. I show how his amalgam of ancient principles completes a reversal of the Thomist order of nature already begun in the works of Piccolomini and Zabarella. Descartes was certainly familiar with Basso’s writings and so this is a

defends the controversial view that Kepler did have the concept of a general, causal law of nature, he argues that his ‘first law does not, in Kepler’s view, have a lawlike character because it offers no connection between the causes of motion and the resulting components of motion’ and that his second law ‘was regarded by Kepler only as a tool with which the motions of the planets could be calculated much more efficiently’ (Grasshof 2008, 160). ¹⁴ According to Van Berkel, ‘Beeckman thus completed his path to the modern principle of inertia in two stages. First he formulated inertia only for the heavenly spheres and with that remained within the Aristotelian framework. Then he rid himself of the Aristotelian framework and declared that his principle was applicable to all motions, natural and unnatural (concepts that lost their significance), superlunary and sublunary.’ He had taken the first step by 1612, but he only worked out the full implications over time (Van Berkel 1983, 192, translation mine). ¹⁵ Van Berkel (1983, 169). ¹⁶ For Beeckman, true causes of natural phenomena are intelligible and sensible and when they are not, we must reason by analogy to things that are. For this reason Beeckman rejects occult causes: ‘According to Beeckman the investigator must always keep clearly in view that there is in nature a fundamental division between matter and spirit. Both domains of nature are so different that one cannot understand how one could apply the concepts from one domain to another. Whoever nevertheless does do it produces occult explanations and unholy confusion. One example is the attribution of intelligence to material objects like planets. Another is the positing of a world soul or “anima mundi” (III, 18). When Kepler talks about a “formative power” in nature, Beeckman rejects it out of hand’ (Van Berkel 1983, 164–5, translation mine). ¹⁷ When citing Aquinas, the following abbreviations are used: ‘CAM,’ Commentary on the Metaphysics of Aristotle (1961); ‘DV,’ De Veritate (1972); ‘LMP,’ On Law, Morality and Politics (1988); ‘QDV,’ Quaestiones Disputatae de Veritate (1992a); ‘SENT,’ In III Sententiarum (1992b); ‘SCG,’ Summa Contra Gentiles (1924); ‘ST,’ Summa Theologica (1988).

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crucial historical as well as conceptual link to Descartes’s recognizably modern way of understanding the order and laws of nature. By the seventeenth century, the Jesuits were the premier educators in Europe. Eager to keep their curriculum up to date, they educated their students about new scientific discoveries. However, the fundamental principles of their natural philosophy still generally followed Aquinas’. It was the explicit policy of the Jesuit order to follow Aquinas’ theology and his interpretation of Aristotle’s philosophy.¹⁸ I first discuss fundamental principles of the Thomist order of nature so as to understand the predominant view taught in many schools across Europe at Descartes’s time. Since my concern is with the origins of Descartes’s concept of a law of nature I focus on Aquinas’ teachings pertaining to the law-likeness of nature. The notion of a universal law governing nature is not foreign to Aquinas and his Jesuit followers but, as I will show, it has a very different meaning from the modern understanding we find in Descartes. Aquinas’ view of the natural order has three notable features, the first two of which stand in stark contrast to Descartes’s. First, it is essentially teleological. There are two ends towards which things are aimed, an extrinsic and an intrinsic end. The ends are good and created things have a natural desire or appetite to reach them. Second, all order within the universe derives from this directedness towards the end. There is a hierarchy of being according to how far removed a thing’s nature is from the ultimate end, and a corresponding order in the relations and causal actions amongst things. Third, prior to the natural order found in created things and their goal-directed action, there is the order which proceeds from the intention of the First Cause. Aquinas also claims that the world is governed by divine providence and the eternal law; moreover, only God himself is able to transcend the laws of nature. According to Aquinas’ teleological view of nature, the end of created things is the common good of the whole universe. Thus the good of every particular thing is directed to the good of the whole. In addition to this intrinsic end there is an extrinsic end towards which the universe as a whole is directed: the Divine Goodness.¹⁹ The whole universe is thus directed towards God and since God is the supreme and most desirable good this directedness involves an appetite on the part of creatures. All things, whether they are rational or irrational, strive to be like God. In nature, this manifests itself in the striving of matter (the potential and less perfect) for form (the actual and more perfect).²⁰ The order found in the universe derives from the directedness of each of its parts to the common good of the whole and the directedness of the whole to God.²¹ So on the one hand, the universe is orderly because it is directed towards the good, but

¹⁸ In fact the ‘Rules for the Professor of Philosophy’ in the Ratio Studiorum of 1599 states: ‘Let him never speak except with respect of Aquinas, following him readily as often as it is proper, or reverently and gravely differing from him if at any time he does not approve of him’ (Donohue 1963). ¹⁹ Wright (1957, 32–3, 36). ²⁰ Marling (1934, 65–6). ²¹ Wright (1957, 32).

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on the other hand, what makes the universe good is its order. There is thus a mutual dependence between good and order and Aquinas goes so far as to say that the good of the universe is the mutual order of its parts.²² Just as Aquinas distinguishes between an intrinsic and an extrinsic end of the universe, he distinguishes between the universal and the particular orders of the universe. Each part of the universe is directed to the common end indirectly by being directed to a specific end such as its own actuality. This results in particular orders, each of which consists in a series ordered to a particular and more immediate end. A universal order is a group of particular orders related to one another by being subordinated to a higher end which subsumes their particular ends. The universal order is thus the sum of all particular natural orders subordinated to the ultimate final end, God.²³ This brings us to the second notable feature of Aquinas’ conception of order, namely, its hierarchical nature. The parts of the universe are ordered into a whole ‘according as one acts on the other, and according as one is the end and exemplar of the other.’²⁴ This is not surprising since being itself is hierarchical in the sense that there is a range of existing things progressing by decreasing increments from the most perfect (God) to the least perfect (matter). Since natural things have forms of differing levels of perfection there must be corresponding differences in their actions. In fact, according to Aquinas, the higher a thing’s form is on the ladder of being, the more unified and extensive its actions will be. Thus the higher natures perfect and rule over the lower by their actions and the lower natures exist for the sake of the higher.²⁵ Here Aquinas incorporates the Neoplatonic idea that goodness overflows from the highest beings to the lowest in decreasing amounts. This hierarchy of actions and influences flowing from the hierarchy of forms is referred to as the order of causes. The division within the order of causes corresponds to the division between three types of being. First there is the everlasting and immutable Being of God. Correspondingly, God is a transcendent cause and exercises a divine and universal causality. The second level of being consists in the incorruptible but changeable heavens. Their causality is universal in that it extends to all change in physical nature, but it is particular in that it is limited to natural motion. Lastly, there are the corruptible things on earth. They are the particular, contingent causes from which contingent effects follow.²⁶ In concrete terms, the universal celestial causes preserve the plants and animals which are for the sake of man, the highest of the earthly creatures.²⁷

²² The order of the distinct parts of the universe is equated with the good and final end of the production of the universe: ‘But the good and the best in the universe consists in the mutual order of its parts, which is impossible without their distinction from one another; for by this order the universe is established in its wholeness, and in this does its optimum good consist. Therefore, it is this very order of the parts of the universe and of their distinction which is the end of the production of the universe (Aquinas 1972, Bk II, ch. 39, 117). ²³ Marling (1934, 39). ²⁴ Aquinas (ST 1a, Pt1, vol. 2, q. 48, a.1, 264). ²⁵ Wright (1957, 104–5). ²⁶ Marling (1934, 113–14). ²⁷ Wright (1957, 144–5).

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As Aquinas writes, ‘According to the order of nature, the active forces of the elements are subordinate to the active powers of the heavenly bodies.’²⁸ So far I have discussed the teleological and hierarchical aspects of the order of nature. Each level of being is directed towards a more perfect form of being and these particular strivings are unified into a universal order by being directed to the ultimate end of the Divine Goodness. From the point of view of the actions of natural things, the order of nature is derived from the upwards striving of things towards the final end. However, God is not only the final cause of the universe but also the first efficient cause whereas causes operating at the second and third levels of being both count as secondary efficient causes. From this point of view, the order found in nature does not arise from nature itself but is imposed on it from the top down. In fact, it is this imposed order which is prior for there would be no order in the actions of natural things without God’s command. In his Commentary on Aristotle’s Metaphysics, Aquinas compares God’s rule over the universe to the rule of a father over his household: And just as the order of the family is imposed by the law and precept of the head of the family, who is the principle of each of the things which are ordered in the household, with a view to carrying out the activities which pertain to the order of the household, in a similar fashion the nature of physical things is the principle by which each of them carries out the activity proper to it in the order of the universe . . . Now the nature of each thing is a kind of inclination implanted in it by the first mover, who directs it to its proper end; and from this it is clear that natural things act for the sake of an end even though they do not know that end, because they acquire their inclination to their end from the first intelligence.²⁹

Furthermore, Aquinas draws on Aristotle’s comparison between an army’s relationship to its commander, and the world’s relationship to the first mover to illustrate the dependence of the order of the universe on God’s will.³⁰ Before it is realized, the order among natural things exists in God’s intention; i.e. it exists as a plan in God’s intellect. One could say that God has many plans since the eternal patterns of finite things exist in God’s mind as Divine Ideas (rationes aeternae) and he freely chooses which to create. The plan or order which is perfect for God’s purpose is realized when God creates the order of creatures and their actions.³¹

²⁸ Aquinas (SCG, vol. III, part 2, ch. 99, 57). ²⁹ Aquinas (CAM, Bk 12, Lesson 12, 2629, 919). ³⁰ ‘And since the formal character of things which exist for the sake of an end is derived from the end, it is therefore necessary not only that the good of the army exist for the sake of its commander, but also that the order of the army depend on the commander, since its order exists for the sake of the commander. In this way too the separate good of the universe, which is the first mover, is a greater good than the good of order which is found in the universe. For the whole order of the universe exists for the sake of the first mover inasmuch as the things contained in the will and mind of the first mover are realized in the ordered universe. Hence the whole order of the universe must depend on the first mover’ (CAM, 2631, 919–20). ³¹ Marling (1934, 72, 86).

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This brings us to the third notable feature of Aquinas’ view. Given that God orders the actions of all things to their end when he creates them, God is said to govern all things. All created things thus fall under the rule of divine providence.³² The analogy between divine providence and human law is established in the following texts: So the world is governed through the providence of that intellect that gave to nature this order, and we may compare the providence through which God governs the world to the domestic foresight through which a man governs his family or to the political foresight through which a ruler governs a city or land, directing the actions of others towards a definite end with respect to himself.³³ The impression of an inward active principle is to natural things what the promulgation of law is to men because law, by being promulgated, imprints on man a directive principle of human actions, as stated above.³⁴

Insofar as it creates all things the divine wisdom contains the eternal exemplars or patterns of all things. As ruler of all created things it is the author of the eternal law to which all things are subject. Wherefore, as the type of the divine wisdom, inasmuch as by it all things are created, has the nature of art, exemplar, or idea, so also the type of divine wisdom, as moving all things to their due end, has the nature of law. Accordingly, the eternal law is nothing else than the type of divine wisdom, as directing all actions and movements.³⁵

In the following passage Aquinas defines God’s law on the basis of a similarity to human law. As stated above, law is nothing other than a certain dictate of the practical reason in the ruler who governs a perfect community. Now it is evident, however, granted that the world is ruled by divine providence, as was stated in the First Part, that the whole community of the universe is governed by divine reason. Wherefore the very idea of the government of things in God the Ruler of the universe has the nature of law. And since the divine reason’s conception of things is not subject to time but is eternal, according to Pr. 8:23, therefore it is that this kind of law must be called eternal.³⁶

It is important to note that the eternal law governs all of nature, irrational and rational creatures alike. Now just as man, by such pronouncement, impresses a kind of inward principle of action on the man that is subject to him, so God imprints on the whole of nature the principles of its proper actions. And so, in this way, God is said to command the whole of nature, according to Ps. 148:6: ‘He has made a decree, and it shall not pass away.’ And thus all actions and movements of the whole of nature are subject to the eternal law. Consequently, irrational creatures are subject to the eternal law, through being moved by divine providence, but not, as rational creatures are, through the understanding of the divine commandment.³⁷

³² Wright (1957, 3–34). ³³ Aquinas (DV, q.5, a.2, 189). ³⁴ Aquinas (LMP, ST 1a2ae, q.93, art. 5, reply objection 1, 41). ³⁶ ST 1a2ae, q.91, a.1, 18. ³⁷ ST 1a2ae, q.93, a.5, 41.

³⁵ ST 1a2ae, q.93, a.1, 34.

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God impresses His eternal law on the natures of things and they carry out its dictates in their actions. In natural things, God’s eternal law translates into their innate tendencies to act in certain ways.³⁸ It is at this level that the expressions ‘lex naturae’ (law of nature) and ‘lex naturalis’ (natural law) are used. The latter phrase is more common and well studied. It refers to the participation of rational creatures in the eternal law and forms the basis for Aquinas’ political philosophy. For Aquinas this form of participation is the more perfect one. Even irrational animals partake in their own way of the eternal reason, just as the rational creature does. But because the rational creature partakes thereof in an intellectual and rational manner, therefore the participation of the eternal law in the rational creature is properly called a law, since a law is something pertaining to reason, as stated above. Irrational creatures, however, do not partake thereof in a rational manner; wherefore there is no participation of the eternal law in them except by way of similitude.³⁹

Since the rational creature participates in providence in a more perfect way and since it is at the peak of the earthly hierarchy, with all other creatures existing for its sake, it is not surprising that Aquinas devotes more time to discussing the natural law governing human actions than he does to the laws of nature governing non-rational things. As the passage makes clear, the governance of natural law over rational beings is the primary sense of a law of nature which is then applied to non-rational creatures by way of analogy.⁴⁰ Although it is definitely rarer, there are contexts in which Aquinas applies the term ‘lex naturae’ to nature as a whole and to non-rational beings specifically.⁴¹ The most common of such usages occurs in contexts where Aquinas contrasts divine action, which is not bound by the laws of nature, with the actions of creatures. Aquinas emphasizes that only God can change the law of nature, which means that every other being, rational or irrational, is bound by it: ‘to him alone it belongs to change the law and course of nature which has been imposed, who established and ordained it, which only God did.’⁴² God being the legislator of the law is the only one who has the power to change it. God changes the law of nature by doing works that transcend or go above the laws he has established. Such works are miracles and are not so much contrary to nature as above it. ³⁸ Marling (1934, 86–7). ³⁹ Aquinas (LMP, ST 1a2ae, q.91, a.2, 20). ⁴⁰ As Marling puts it, ‘It is only with regard to rational creatures, therefore, that the term law is used in its strict and proper meaning. If it is applied to the irrational world it is merely by way of analogy. To include this latter case law may be defined in a wide sense: “lex importat rationem quamdam directivam actuum ad finem.”’ (Marling 1934, 83). ⁴¹ Though it is rare in Aquinas, Marling points out that St Augustine speaks of laws of nature with reference to the physical universe and does not use the term merely in an analogical sense. Bodies participate in the eternal law given by the Divine Wisdom and Will no less than rational creatures do (1934, 77–8). Similarly, Aquinas’s contemporaries do not hesitate to apply the term ‘lex naturae’ to material creatures as seen in In Boethii de Consolationes Philosophiae by Guillelmus Wheatley and Postilla in Librum Geneseos by Petrus Johannis Olivi (CD-ROM, Roberto Busa). ⁴² Aquinas (Sent. DS. 16, q.1, a.3).

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I answer that, Since every creature is subject to the laws of nature, from the very fact that its power and action are limited: that which surpasses nature, cannot be done by the power of any creature. Consequently if anything needs to be done that is above nature, it is done by God immediately; such as raising the dead to life, restoring sight to the blind and suchlike.⁴³

The above passage makes it clear that the laws of nature apply to limited beings. One could go so far as to say that the laws seem to be restrictions which serve to delimit the actions of creatures. Only God, being unlimited, is not subject to them. It appears from this that the laws of nature are closely connected to the natures of creatures and their characteristic ways of acting. Perhaps the laws of nature express the fact that each creature is restricted by its form to a limited range of action. However, the following passage indicates that the law of nature is something over and above the actions which spring from a creature’s form. Rather, it seems to be an expression of divine power: ‘and therefore it is not in the power of a spiritual substance to be united or separated from a body by way of the form, but this is done by law of nature or divine power.’⁴⁴ However, in the passage below, natural law is once again tied to the inclinations springing from a creature’s nature. It is also clear that although Aquinas focuses on the natural law as it applies to humans, non-rational beings likewise have inclinations which fall under the natural law. Because in man there is first of all an inclination to good in accordance with the nature which he has in common with all substances, inasmuch as every substance seeks the preservation of its own being according to its nature, and by reason of this inclination, whatever is a means of preserving human life and of warding off obstacles belongs to the natural law. Second, there is in man an inclination to things that pertain to him more specifically according to that nature which he has in common with other animals, and in virtue of this inclination, those things are said to belong to the natural law ‘which nature has taught to all animals’ such as sexual intercourse, education of offspring, and so forth.⁴⁵

Descartes’s natural philosophy is at odds with all three notable features of Aquinas’ view of the natural order, most obviously, the first two. Since he denies that we can know God’s ends, Descartes rejects the appeal to final causes, and thereby the corresponding causal hierarchies, in the study of nature.⁴⁶ When he characterizes the laws of nature as ‘secondary causes’ of the motions of particular things, he cannot

⁴³ Aquinas (ST 1a2ae, q.5, a.6, 80). ⁴⁴ Aquinas (QDV, 2 q.26, a.1). ⁴⁵ Aquinas (LMP, ST 1a2ae, q.94, a.2, 48). ⁴⁶ ‘When dealing with natural things we will, then, never derive any explanations from the purposes which God or nature may have had in view when creating them . For we should not be so arrogant as to suppose that we can share in God’s plans. We should, instead, consider him as the efficient cause of all things; and starting from the divine attributes which by God’s will we have some knowledge of, we shall see, with the aid of our Godgiven natural light, what conclusions should be drawn concerning those effects which are apparent to our senses’ (CSM I, 202).

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thus mean exactly what Aquinas does. However, by the late sixteenth century, influential Jesuit philosophers had redefined the causal role of secondary causes and chipped away at final causes to such an extent that their conceptions of secondary and final causation are much closer to Descartes’s.⁴⁷ As a result, some share with Descartes an emphasis on material and efficient causation to explain natural phenomena.⁴⁸ Yet none of Descartes’s Jesuit teachers were thereby led to abandon the broader Aristotelian picture of the natural world as populated by individual substances characterized by their natural powers and inclinations. One might conclude that Descartes, having redefined matter as passive res extensa, must reject the teleological and hierarchical features of an Aristotelian natural philosophy in favor of locating the natural order entirely in God’s governance. However, contemporaneous proponents of atomist or corpuscularian matter theories are not necessarily led thereby to see all nature as governed by divine laws.⁴⁹ Therefore, Descartes’s new matter theory and consequent redefinition of secondary causation and rejection of Aristotelian teleology cannot alone account for his universal, causal, and determinative conception of the laws of nature. The third feature of Aquinas’ view, that there is a prior, universal order from the ideas and exemplars in God’s mind, some of which God chooses to realize when he freely creates the world, at first appears to be close to Descartes’s. This order impressed on God’s creation constitutes the divine providence which governs nature, and Aquinas refers to it as natural law, or law of nature, by explicit analogy to the law that governs human society. Aquinas’ concept of a universal law, of divine origin, which plays a causal role in governing nature might appear as a likely source of Descartes’s view that all of matter and its motions are determined by the three divinely grounded laws of nature. However, as Jean-Robert Armogathe has shown, Jesuit philosophers significantly revised Aquinas’ view of eternal law, incorporating elements of Scotism.⁵⁰ Armogathe highlights that the overarching problem with Aquinas’ view, which motivates disputes among sixteenth- and seventeenth-century theologians, is that if natural law constitutes the expression of divine wisdom then the intelligibility of the world is privileged over divine omnipotence.⁵¹ Jesuit theologians modified Aquinas’ view in various ways to avoid this result. ⁴⁷ Final causes were increasingly regarded as causes in a metaphorical sense and the causal role of secondary efficient causes is also reduced. One influential Jesuit philosopher’s arguments are examined in detail by Akerlund (2011). On late Scholastic views of final and efficient causation see also Des Chene (1996). On late Scholastic conceptions of the respective contributions of secondary and divine causation in producing an effect see Freddoso (1991, 1994) and Hattab (2003, 2007). ⁴⁸ See Hattab (1998, 2012). ⁴⁹ The atomist natural philosophies developed by David Gorlaeus and Pierre Gassendi do not include laws of nature. ⁵⁰ After Aquinas’s view was targeted in the 1277 Condemnation, Franciscans, like Scotus, defended divine omnipotence and the primacy of divine will over intellect. In the seventeenth century, opponents of Thomism rejected the view that the eternal law in God is the same as the reason for government of nature in God (Armogathe 2008, 266–8). ⁵¹ Armogathe (2008, 266–7).

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Gabriel Vasquez criticizes the Thomists for trying to give the eternal law the characteristics of a law. Rather he recommends the term ‘law’ be abandoned in favor of the term ‘idea’ which is a type of law but one that is neither prescriptive nor in need of promulgation.⁵² This serves to create enough of a gap between the natural order and God to avoid subjecting God’s eternal decree to the intelligible order of nature. Francisco Suarez, Vasquez’s contemporary and rival, takes a different route in his De Legibus. With Aquinas, he affirms that eternal law is both eternal and a law but qualifies that it is so in two different respects. It is eternal with respect to its status as an internal disposition in God, the legislator, and it is a law with respect to its status as constituted and promulgated outside the legislator’s thought. The law which applies to creatures is thus not eternal and so does not compromise divine omnipotence by subjecting God’s will to the law of nature.⁵³ Suarez further emphasizes the metaphorical meaning of ‘law of nature’ with respect to inanimate things.⁵⁴ According to Armogathe, Suarez then, drawing on Plato, makes an original distinction between a moral law, proposed to humans, and the law of the mechanics (lex artificiorum) which presides over creation. The latter does not require government and promulgation and so law only properly applies to morality.⁵⁵ With Vasquez’s and Suarez’s revisions to the Thomist view, we could not be further from Descartes’s concept of an eternal law. Descartes’s three laws of motion are neither ideas in the divine mind nor applicable to inanimate things only metaphorically—to the contrary, Descartes’s laws follow from God’s immutability and apply only to motions, which are modes of material substance not of thinking substance. Since his view does not conform to that of his Jesuit teachers, one might instead see an affinity between Descartes’s concept of a law of nature and voluntarist theological doctrines.⁵⁶ Descartes holds a radically voluntarist view with respect to the eternal truths, claiming that they are freely created by God’s efficient causality and are true because God willed them, not vice versa; hence God could have created a different set of eternal truths. For Descartes, the eternal truths are first and foremost the truths of mathematics, so this could imply that the mathematizable rules derived from the third law, which governs collisions between bodies, could have been entirely different. However, since Descartes claims to derive the three laws of motion directly from God’s immutability, it is more plausible that unlike the eternal truths, these laws are not products of God’s will.⁵⁷ On this reading, Descartes’s laws are not created ideas, ⁵² Armogathe (2008, 269). ⁵³ Armogathe (2008, 273–4). ⁵⁴ Armogathe (2008, 272). ⁵⁵ Armogathe (2008, 273–4). ⁵⁶ This background to the modern concept of ‘law of nature’ is discussed in Oakley (1961) and used to account for differences between Descartes’s and Gassendi’s views on the relationship between God and nature by Osler (1994). Roux (2008) explores a similar theological debate in Descartes’s successors and how it informs their conceptions of the laws of nature. ⁵⁷ This inference presupposes that Descartes conceives of divine acts of will as acts which could have been performed otherwise. This might appear to contradict his claim in Meditation 4 that ‘In order to be free, there is no need for me to be inclined both ways; on the contrary, the more I incline in one direction— either because I clearly understand that reasons of truth and goodness point in that way, or because of a

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distinct from God’s essence, but rather expressions of God’s immutability and thus essentially identical with God.⁵⁸ But if that is the case, then they are very different from the eternal decrees of the divine will which for voluntarist theologians constitute the eternal law. Regardless of which interpretation one takes, the question of how the three laws of motion relate to God’s nature is tangential to our search for conceptual roots of Descartes’s laws of nature for this is not the level at which the term ‘law of nature’ applies, whether one is a Thomist, Scotist, or Jesuit theologian. In other words, it is not so much the eternal law of the divine will/mind that is relevant, but the order that gets encoded in the natural inclinations of things. This is the level that corresponds to Descartes’s conception of the laws of nature as secondary causes of particular motions. In looking for the philosophical roots of Descartes’s concept, we must thus shift our focus to the conceptual problem space at this level as found in works on natural philosophy. Insofar as sixteenth- and seventeenth-century Scholastic works on physics treat the way in which divine providence is impressed on nature, they often address this issue in the context of the question regarding whether fate is a cause. The works of influential natural philosophers, Domingo de Soto, Francesco Piccolomini, and Jacopo Zabarella, are fairly representative and shed light on the philosophical issues at stake and how these were addressed. Domingo de Soto in his 1583 Commentary on the Eight Books of Aristotle’s Physics is primarily concerned to affirm divine providence without attributing irregularities in nature (such as the birth of monsters) to God’s causality and subjecting human will to fate, fortune, or chance. Soto cites Ancients who report that the Stoics defined fate as ‘the series and order of natural causes, which produce the effect by inevitable necessity’ and rejects both the Stoic view as well as Cicero’s denial of divine foreknowledge.⁵⁹ Soto denies ‘fate to be a cause acting necessarily and inevitably’ and affirms ‘that his [God’s] foreknowledge imposed no necessity on things.’⁶⁰ God’s ‘concourse is governance and providence which does nothing contingently and indeterminately but does all things from a certain judgment.’⁶¹ However, this does not imply that necessity is imposed on things for ‘fate is in fact a cause per se inclining towards some effect: which nevertheless can be impeded through another particular cause.’⁶² divinely produced disposition of my inmost thoughts—the freer is my choice’ (CSM II, 40). Descartes, in this context, discusses the human will. In a letter to Mersenne that discusses God’s creation of the eternal truths Descartes takes God’s freely willing these truths to involve God’s ability to will (and understand) them otherwise: ‘You ask also what necessitated God to create these truths; and I say that he was just as free to make it not true that all the lines drawn from the center to the circumference were equal as he was to not create the world. And it is certain that these truths are no more necessarily conjoined to his essence than the other creatures. You ask what God did to produce them. I say that he created them from the very same [act] that he willed and understood them to be from eternity’ (AT I, 150, translation mine). God’s immutability, by contrast, is joined to his essence. ⁵⁸ Pavelich (1997) and Garber (2013) argue for this interpretation. ⁶⁰ Soto (1583, 59). ⁶¹ Soto (1583, 59). ⁶² Soto (1583, 59).

⁵⁹ Soto (1583, 59).

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Soto accomplishes this result by restricting fate to the level where God’s providence is translated into natural causation: ‘fate is not God, nor his providence, but the order of natural things about to be prescribed.’⁶³ Fate is ‘the order of things, preordered and premade by God’ but to avoid necessitarianism, Soto limits the operation of fate (and also chance and fortune) to effects that depend on God’s concurrence with secondary causes.⁶⁴ Things that are perpetual by their own nature, like the creation of the world, the conservation of angels and the celestial orbs, depend on God alone, and so lie outside the realm of fate. The human will is likewise outside its scope as ‘in its actions it depends immediately on God alone.’⁶⁵ Fate thus designates only the natural order of secondary causes. These are naturally inclined to produce specific effects but can be thwarted, e.g., by particular causes impeding the proper disposition of matter such that a monster is born. Soto does not use the term ‘law of nature’ but his concept of fate as the order of natural things to be prescribed by God plays an analogous role to Descartes’s three laws of motion. This order, once prescribed, can lead to irregular results due to impeding particular causes altering the dispositions of matter. Descartes similarly identifies the laws of nature as ‘secondary causes,’ and attributes irregularities in motion to the dispositions of matter, ‘Thus following this rule,⁶⁶ it must be said that God alone is the author of all the motions that exist in the world in so far as they exist and in so far as they are straight; but it is the diverse dispositions of the matter which render them irregular and curved.’⁶⁷ But despite the analogy, there is a crucial difference between Soto’s and Descartes’s views. For Soto, the order of nature is identified with the inclinations of individual natural things to act towards their natural ends. When particular causes with opposing inclinations intervene, the activity of a natural substance fails to realize its natural end, producing an unnatural result, like a monster. For Descartes, by contrast, the natural order is not defined in terms of individual substances with intrinsic inclinations to act towards certain goals. In Meditation 6 he compares the human body to a clock. Just as ‘a clock constructed with wheels and weights observes all the laws of nature just as closely when it is badly made and tells the wrong time as when it completely fulfills the wishes of the clockmaker’ so the body suffering from dropsy does not depart from its nature except in an extrinsic sense, imposed by human thought.⁶⁸ It is nature in the intrinsic proper sense, namely nature as conformity to the laws of nature, that for Descartes refers to ‘something which is really to be found in the things themselves.’⁶⁹ Hence for him, there are no unnatural effects—the processes that produce monsters conform to the laws of motion just as all natural processes do.

⁶³ Soto (1583, 59). ⁶⁴ Soto (1583, 59). ⁶⁵ Soto (1583, 59). ⁶⁶ In The World, this is the third rule or law of motion that the parts of a moving body individually always tend to continue moving along a straight line, whereas in the Principles of Philosophy this becomes the second law of motion. ⁶⁷ AT XI, 46–7. ⁶⁸ CSM II, 58. ⁶⁹ CSM II, 59.

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It is not just that Descartes has a different account of matter from Soto, but in addition, between Soto and Descartes a reversal of Aquinas’ order of nature has taken place. For Soto, as for Aquinas, the order of nature, which he calls ‘fate’ instead of ‘law,’ is still dependent on regular patterns of activity rooted in the inclination of each natural substance to act towards its natural end.⁷⁰ Hence, even though prescribed by God, the order impressed on creation can lead to divergences from what is natural to individual substances when their natures are prevented from reaching those ends by other elements of the causal nexus. This is precluded on Descartes’s view and it is not merely because he rejects teleological explanations in favor of mechanical ones on the grounds that God’s aims are unknown to us. His rejection of final causes in natural philosophy is only part of the bigger story. Descartes also holds that what we ordinarily attribute to the ‘nature’ of an individual substance is an extrinsic attribution, a mental construct that fails to correspond to what is found in things themselves. Thus, Descartes denies the Scholastic view that individual substances really have natures in the Aristotelian sense of a substantial form that accounts, e.g., for the normal biological activities of the healthy body versus that of the patient suffering from dropsy. Aristotelian substances and their individual natures are not metaphysically prior for Descartes as they are for Soto. On Descartes’s metaphysics, the laws of nature are prior and the patterns of activity (first and foremost, the lawlike motions of material particles) depend on these laws, not on individual substances in Aristotle’s sense. Nature is then redefined as conformity to these laws with the result that no motion diverges from the natural order. Descartes’s concept of a law of nature is thus part and parcel of a broader metaphysical reconceptualization of nature and its order. By regarding individual substances and their true natures as dependent on underlying laws and the motions that follow, Descartes fundamentally revises Aristotle’s Categories, according to which, accidents, like motion, only exist in and always depend on primary substances. It is clear now that, due to his rejection of the Aristotelian ontological order of dependence, Descartes’s concept of a law of nature, despite superficial terminological and functional resemblance, is radically different from what we find in his Scholastic predecessors. Due to deeper differences at the metaphysical level it cannot be traced back to Scholastic Aristotelian conceptions of divine providence, natural law, and fate. Hence we cannot account for Descartes’s concept of a law of nature by supposing that his innovation consists in having rules of motion that resemble the principles of Beeckman’s physics, occupying the place that for Scholastic Aristotelians is occupied by the natural law of divine providence. However, this does not mean that Descartes’s commitment to the metaphysical priority of the laws of nature is entirely without precedent. Neoplatonic principles that make their way into

⁷⁰ As John R. Milton highlights, Richard Hooker does use the term ‘law’ for essentially the same concept in his Laws of Ecclesiastical Polity of 1593 (Milton 1998, 2).

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Piccolomini’s and Zabarella’s natural philosophies, via Averroes and Alexander of Aphrodisias, and, to a greater extent, Neoplatonic and Stoic elements of Basso’s explicitly anti-Aristotelian work on natural philosophy, give rise to conceptions of the order of nature and its laws closer to Descartes’s. Piccolomini and Basso posit a universal causal principle that is prior to and independent from individual substances and the source of the lawlike order in nature. Francesco Piccolomini’s Librorum ad scientiam de natura attinentium pars 1–5, first published in 1596, includes a ‘Digression Pertaining to Nature’ in the Second Book. The third part of the digression explores the connection between nature and fate. The first part recounts the different senses of ‘nature,’ of which the eleventh is ‘the series and order of general causes, which is called the general Nature and by the Poets is called the God Pan, specifying the whole and the universe; and to him they attribute the flute marking the consonance of the universe and many other things according with the conditions of the universe.’⁷¹ Right before he addresses the nature of fate, Piccolomini divides nature into natura naturans and natura naturata, taking up a distinction he traces back to Boethius and Averroes. Piccolomini attributes to Plato the opinion that natura naturans is the World Soul pouring the seeds into matter, whereas natura naturata is matter endowed with seeds.⁷² He attributes to Aristotle a threefold sense of natura naturans based on three comparisons: 1) God compared to the higher minds (i.e., the intelligences) is natura naturans because he is the origin of all things and the other minds, since they are dependent on God, are natura naturata, 2) the Heavens when compared with mortals are natura naturans and the seeds pouring in an arrangement under it natura naturata, 3) form and matter when compared to the composite are natura naturans and the composite natura naturata. Piccolomini then adds a fourth sense to Aristotle’s: ‘the universal nature taken up on behalf of the series of causes is natura naturans, however the things coming forth from it are natura naturata.’⁷³ In fact, in Piccolomini’s previous discussion of the three senses of the distinction between universal and particular nature, the first two exactly mirror the first two senses of natura naturans versus natura naturata. The third and most proper sense differs in that the universal Nature is not matter and form in comparison to the composite but rather ‘a series of causes, ordered by God and the superior minds, descending towards the Celestial bodies, and thirdly towards the Elements, and progressing in this way all the way up to the final effect.’⁷⁴ Having clarified these key distinctions, Piccolomini quickly connects this universal nature to fate, claiming that fate is completely joined with nature ‘especially that which is called universal and general.’⁷⁵ Piccolomini attributes to Aristotle the view that fate is the same as nature but conceptually distinguished from it in that ‘nature’ refers to the essence of a thing, and ‘fate’ to ‘the order depending on a certain principle, ⁷¹ Piccolomini (1600, 118v). ⁷⁴ Piccolomini (1600, 118).

⁷² Piccolomini (1600, 118r). ⁷⁵ Piccolomini (1600, 118).

⁷³ Piccolomini (1600, 118).

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just as predicted, commanded and revealed.’⁷⁶ He then combines the Neoplatonic seed metaphor with the legislative analogy we encountered in Aquinas: Fate in fact sows beforehand the order depending on a certain principle, just as predicted, commanded and revealed. For as in the Civil rule, which follows the constitution, promulgation and execution of the law, he is considered the leader with whom wisdom and Providence is congruent. Thus in the universe, the leader of all things is God, and Providence follows the divine mind, to which the most eminent wisdom belongs, just like the promulgation of Law [follows the constitution of law]. Fate follows nature, just like the execution of the Law.⁷⁷

Piccolomini concludes, ‘Henceforth, nature, as following fate, is called ordered; in itself it is order; and related to the things that follow, it is called the proximate cause of the order.’⁷⁸ Finally, since fate is only conceptually distinct from nature, it too is divided into universal fate which is the promulgation in the universe of the law of the order of the universe, and particular fate, which is contracted to particular things and called fate only because it depends on and is joined to universal fate. Properly speaking and absolutely, fate is universal fate. Here we have the beginnings of the ontological reversal that will become more pronounced in Basso and Descartes. Fate, defined as the series and order of general causes, is not simply the regular patterns of all unthwarted causal activities of particular natures taken together, as in Soto. It is rather natura naturans, a universal nature/essence, which is prior to the inclinations of particular natures, and brings forth universal fate or the general order of causes (natura naturata). This implies that the universal nature as active source of the entire natural causal order is metaphysically prior to it. Piccolomini refers to some authors (without naming them) who divide it into Divine fate, which comes from God, Astronomical fate, which comes from heavenly bodies, and Physical fate. Piccolomini, like Soto, defines fate as the order prescribed to nature and he also supposes that in addition to the universal nature, there are particular essences that contract its universal causality to particular ends. However, as seen, he begins to reverse the order of dependence found in Soto. For Piccolomini, the order insofar as it lies in the natures of particular things is not fate properly speaking. Rather, the particular causal order is called fate only because it is joined to and depends on a more fundamental universal order. This more fundamental order is likened to the promulgation of the law in civil society, implying that the natural causal order is merely the execution of the universal order. This suggests that Piccolomini, unlike the Jesuits and Soto, but like Basso and Descartes, regards the natural order of particular causes as determined by the universal order. Whereas Piccolomini does not draw out this implication, one can see how the legislative analogy, if taken literally and combined with the division of fate into divine, astronomical, and physical fate, could promote the search for fundamental laws governing the series of causes in each of these domains. ⁷⁶ Piccolomini (1600, 118).

⁷⁷ Piccolomini (1600, 118).

⁷⁸ Piccolomini (1600, 118).

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There is at least one instance where the laws of universal nature, in Piccolomini’s sense, are invoked to address a particular phenomenon in physics. Zabarella, Piccolomini’s colleague at the University of Padua, addresses the case of putrefaction in Book II of his De Rebus Naturalibus, which deals with the generation and corruption of mixed things. In the context of addressing the concern that putrefaction is against nature since it leads to the destruction of nature Zabarella writes: the proper nature of mixed things must not be considered here, but rather the universal nature, which is nothing else than the order of all things, or of all causes disposed to a certain order with a dependence on one first principle, so that for individual things, some proper laws are established that cannot be eluded. Therefore, the proper nature of mixed things refuses to putrefy, or to perish in any way, but the universal nature determines that it is submissive to death, and finally to perishing. Aristotle says in the beginning of Bk I on Generation and Corruption that according to this consideration, death is natural, that is, according to the laws of universal nature, which establish that all created things will at some time perish.⁷⁹

Kusukawa documents that some early seventeenth-century Protestant textbook writers took up this concept of universal nature and the associated concept of law in connection with their deep interest in divine providence. In particular, Rudolph Goclenius, a Calvinist philosopher, drew specifically on Zabarella’s account of putrefaction. According to Kusukawa, Goclenius applied Zabarella’s distinction between universal and particular natures to monsters: monsters were not beyond universal nature; only beyond particular nature. Universal nature, following Zabarella, was called ‘lex’ by Goclenius; he then equated universal and particular nature with God’s universal and special providence, and then called the law of nature ‘fatal.’⁸⁰

Via Calvinist philosophers, the concept of universal nature and its law found in Piccolomini’s and Zabarella’s natural philosophies would have been part of the philosophical discourse in the Dutch republic, where Descartes chose to live and philosophize as of the late 1620s. However, ‘law’ in this sense seems to designate a general underlying order on which particular causal series depend rather than a literal law and it still subsumes, rather than obliterates, the individual Aristotelian natures Descartes deems to be mental constructs. The concrete application we find in Zabarella is still exceedingly broad, stating only that ‘all created things will at some time perish.’ We thus have the universality that also characterizes Descartes’s laws of nature, but Zabarella’s law in no way specifies how things will perish whereas Descartes’s laws of motion determine the direction and speed with which body will move upon collision. Aristotelian natural philosophers still relied primarily on individual natures and their inclinations to ground regular patterns of activity (what Zabarella calls ‘proper laws’) to account for particular effects. Universal nature

⁷⁹ Zabarella (1590, 618).

⁸⁰ Kusukawa (2008, 119).

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and its fate/law may have ontological priority on Piccolomini’s quasi-Neoplatonic metaphysical scheme, but the need to appeal to particular natures and their proper laws for causal explanations of specific natural effects remains. On Descartes’s view, as seen, particular natures, such as the body suffering from dropsy, disappear into universal nature such that the three laws of motion impressed upon the material conditions are supposed to do all the work. While this sets Descartes apart from his Scholastic Aristotelian contemporaries, there is a precedent for his rejection of Aristotelian particular natures and their inclinations in the eclectic anti-Aristotelian natural philosophy developed by the Calvinist philosopher, Basso. Basso’s Philosophiae Naturalis Adversus Aristotelem, first published in Geneva in 1621, and again by Elzevier in 1649, has the express purpose of reinstating the physics of the Ancients.⁸¹ It was quickly recognized as an important, albeit flawed work by natural philosophers and Descartes also refers to it in a couple of letters. ⁸² According to Basso, God created microscopic atoms at the beginning of time.⁸³ Aside from God’s power to destroy them, these simple, homogeneous bodies, each possessing a particular property, are indestructible.⁸⁴ Their properties are inalienable and persist when an atom enters into a compound.⁸⁵ There are four kinds of elementary atoms corresponding to the traditional elements of fire, air, water, and earth. Basso adopts the Platonic view that the elements have specific shapes without specifying these shapes, except in the case of fire atoms, which he claims all the Ancients described as sharp and pointed.⁸⁶ The four elements do not possess their own motive forces but rather are moved by the ether. Furthermore, all natural change is due to the local motion of atoms being pushed by the ether.⁸⁷ If one sets aside the fact that the ether or common spirit is the active causal force that moves bodies, rather than God setting them in motion directly, Basso’s view resembles occasionalism. For Basso, material atoms are passive, the motive force coming from the ether, which he characterizes as the instrument of the divine mind. The ether or spirit is a very tenuous corporeal substance that ⁸¹ However, Basso by no means advocates an uncritical approach, stating, ‘we will demonstrate them only by the most certain and evident reasons’ since ‘If the principles of Aristotle and many others which are joined are venerated just like an oracle, they would not only be uncertain, but also supported by no firm reasons’ (Basso 1621, 12). ⁸² Descartes was familiar with Basso’s work. In a letter to Mersenne dated October 8, 1629 he writes: ‘As for rarefaction, I am in agreement with this physician and have now taken a position on all the foundations of philosophy; but perhaps I do not explain the ether as he does’ (AT I, 25). The physician in question, formerly thought to be Villiers, is in fact Basso (AT I, 665). A year later, on October 17, Descartes lists Basso as one of the novatores who has nothing to teach him in an acrimonious letter to Beeckman (AT I, 158). ⁸³ Basso (1621, 14). ⁸⁴ Basso (1621, 125–6). ⁸⁵ Basso (1621, 73–4). ⁸⁶ Basso (1621, 109). ⁸⁷ Basso (1621, 387–8). Beeckman likewise relies on the ether to explain many natural phenomena but equates it with the fire particles. Van Berkel notes, ‘With some good will one could even say that Beeckman’s views on the construction of the world evolved more and more in the direction of an “ether theory,” a theory in which the world ether, an omnipresent, very rarefied and all-penetrating substance was the dominant element’ (Van Berkel 1983, 178).

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pervades the universe, entering in-between the parts of matter. For example, during rarefaction it enters in-between the parts of air and separates them. Basso credits the Stoics with this view and states: Behold the Stoics established this clearly for you since they recognized most wisely the common spirit in all elements and things. The spirit in all things would indeed be the same as the wind emitted from the bellows in an organ from which they let off sound. Thus from that spirit entering, all things are moved inasmuch as their aptitude allows.⁸⁸

This Stoic spirit, like the wind, creeps into material things giving them action and motion to the extent that their aptitudes allow. Basso uses the same metaphor of the wind entering the bellows later on when he compares the spirit to air entering the compressed bellows, giving it the form and figure that its structure requires.⁸⁹ So the spirit not only moves and activates the matter but it gives it its form and shape. The spirit cannot give the matter any form it wants but rather the matter has innate aptitudes which enable it to take on a certain structure and form but not others. Basso indicates that the spirit adapts its motion to the matter at hand: If the spirit underlies the matter of fire, I say that it separates and moves those most minute barbs with a motion as quick as the condition of their nature demands. Now if it enters in this way the matter whether of air, of water, or of earth, it gives to it the appropriate measure, inasmuch as the parts require, removing some parts from others, and it moves each one inasmuch as it is ordered.⁹⁰

There is an inherent order in the different types of matter, and the spirit, when it penetrates and separates the parts of a substance, adapts its motion to the order required by the matter. The spirit, Basso explains, readily takes on any form and acts as the proximate and universal instrument of the divine mind, giving all things the motion owed to them.⁹¹ Matter, by contrast, is passive; it does not move unless it is moved by the spirit. All generation, corruption, growth, and decay is explained by the various proportions and arrangements of atoms within a compound.⁹² As for Descartes, what we take to be the natures of individual substances result from the lawlike ways in which their material particles are moved. Whereas Aristotelians like Piccolomini and Zabarella still appealed to the natures of individual Aristotelian substances to do much of the explanatory work, in Basso’s theory we have a clear precedent for Descartes’s view that such substances not only depend on a universal order of motions for their activities, but their very natures/forms derive from it. Basso, however, does not avoid the Aristotelian appeal to final causes as scrupulously as Descartes. He states repeatedly that God created matter so that it would be moved according to certain patterns and laws and that God never moves material particles against their aptitudes to be moved towards their natural places.⁹³ ‘Nor indeed does

⁸⁸ Basso (1621, 333). ⁹¹ Basso (1621, 334).

⁸⁹ Basso (1621, 335). ⁹² Basso (1621, 260–1).

⁹⁰ Basso (1621, 333–4). ⁹³ Basso (1621, 341).

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God act through them otherwise than if they themselves directed their proper impetus into every end by their innate power; nor does he cease to move them more than they would cease if they were moved through themselves.’⁹⁴ Basso appears to attribute passive dispositions to be moved in certain ways to material particles and indicates that God binds himself to act in accordance with them once he has created the material world. In other words, God moves the elements in accordance with their natural places and moves composites in accordance with the proportion and distribution of elements that make them up. Basso often refers to material things as ‘instruments’ of the ether but states that, for the purpose of easier comprehension, he will suppose that things act by themselves when, in reality, God acts through them.⁹⁵ Basso goes on to explain how the more general end of the compound is reached by the elements pursuing their particular goals. It seems that for the purpose of explaining particular phenomena he is willing to speak as though material things pursue their ends by their own powers even though, strictly speaking, they do not. Basso also differs from Descartes in positing a causal intermediary between God and matter. Whereas there is general agreement that Basso regards the four elements as passive instruments, it is unclear that he regards the ether in the same way. Basso, following his assumption that there is a consensus among the Ancients, goes on to equate the Stoic ether with Plato’s World Soul as presented in the Timaeus. He attributes to Plato the view that the soul of the world is fire, fire being the most tenuous substance and the most apt to be used by the First Cause for the purpose of moving the world.⁹⁶ Plato does not identify the World Soul with fire in the Timaeus so Basso’s source is probably a contemporary Platonist. Basso then specifies that the formal account of the World Soul does not consist in fire but ‘in the most diverse, nevertheless most certain proportion of motion, which one ought to observe in the parts of the world. Therefore this account of soul consists in a certain proportion.’⁹⁷ Whereas the formal account of the World Soul consists in a certain proportion of motion for Basso, it also has the power by which other material things are moved. I respond that there is the same difference between Nature and the soul of the world that there is between the instrument and the hand that moves it. Indeed a certain aptitude is required in the instruments: by contrast the power which acts on the instrument is the hand, just as its aptitude brings forth. The Nature of things consists in such a quasi-instrumental aptitude which consequently is diverse in diverse things. Indeed the ratio of the soul of the world is placed in that force such that, like the hand of the most wise mind, it impels each thing to be moved the way it has been constituted. Therefore, that proportion of motion pertains both to nature and to the soul of the world. To the soul of the world insofar as it effects this motion.⁹⁸

The World Soul contains the order of the universe, i.e., the proportions of motion which can be expressed by certain rules of laws or nature as Basso calls them.

⁹⁴ Basso (1621, 315). ⁹⁷ Basso (1621, 340).

⁹⁵ Basso (1621, 317). ⁹⁸ Basso (1621, 341).

⁹⁶ Basso (1621, 307).

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These laws are qualitative; for example, it is a law that earth moves towards the center. The World Soul realizes the order captured by the laws in other material things presumably by giving them form and structure (by entering in-between their parts as air enters a bellows). The proportion of motion, therefore, can be said to belong to nature, in the sense that natural things have an aptitude to be moved in certain ways and such motions are owed to them. However, the force of motion belongs to the World Soul as it impels things to move so, in this latter sense, the proportion of motion also belongs to the World Soul. Basso combines the Stoic ‘pneuma’ or ether with the Neoplatonic spirit or ‘World Soul’ to arrive at a new causal principle. From the Stoics he takes the idea of a material, tenuous substance which moves and shapes matter by pervading it as the soul pervades the body. From Neoplatonism, he takes the idea of an intermediate principle, which links the divine purpose to nature by containing the order and proportion which natural phenomena are to follow. The result is a principle that at once explains the rational order of the universe and gives it a physical embodiment. By equating the Stoic ether with the World Soul, Basso has a physical agent with real powers by which he can explain change in the world. Though the common spirit itself may be mysterious, there is nothing mysterious about its motions. Since it creeps in-between material particles there is no action at a distance. All causality is by impact and at the physical level everything can be explained in terms of efficient causality. Basso concludes with the argument that his theory is far superior to the Aristotelian theory of form. He cites the Aristotelian principle that ‘Nature does nothing in vain’ and claims that they violate their own principle by positing the creation and destruction of so many individual forms. Basso considers his explanation, which he attributes to the Ancients, to be much simpler. Instead of holding that new things constantly come into existence, he explains everything in terms of the motion of the common spirit from one part into another. As Basso writes, Of course in this the immense wisdom of the Founder shines, because it has constituted such an order, and such a proportion of things, that from this order and proportion, as long as here one spirit moves certain parts of the most diverse matter, to the extent that their nature demands, by one local motion it brings forth such diverse and such wonderful changes of things.⁹⁹

Rather than positing the coming into existence and destruction of innumerable qualities, Basso explains the appearance of different qualities as the varied action of the same spirit. Rather than relying on all the different types of motions arising from different types of forms, Basso boasts that he can explain everything in terms of one local motion initiated by one common spirit.¹⁰⁰ He concludes with the words:

⁹⁹ Basso (1621, 343).

¹⁰⁰ Basso (1621, 343).

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We are certain, I say, that with this one local motion, we shall be able to render the natural causes far more easily, certainly and clearly than the Peripatetics do by the creation of infinite qualities and substantive forms in which, since they must continuously and singly be created, the Author of nature is incessantly busy.¹⁰¹

In Basso’s natural philosophy, the individual substantial forms that constitute the natures of individual substances and determine their causal actions in accordance with their natural ends have been eradicated. Instead, there is one universal local motion by which all matter is moved in accordance with the order and proportion constituted by God, the creator. This order and proportion of the divine mind is executed by the World Soul/ether, which enacts the proportions of motion in matter, according to its dispositions. Consistent with the tradition, it is at this level that Basso uses the terms ‘law of nature’ or ‘rule of nature.’ Basso still thinks of these laws in Aristotelian terms, as seen by his example of the law that earth moves towards the center. However, the regular motion of earth is not, as it is for Scholastic Aristotelians, due to intrinsic inclinations stemming from the individual nature of this element, which contracts and delimits the action of the universal nature/cause. Rather it is the direct result of the ether which pushes earth particles downward, executing the divine law that the World Soul impresses on nature. The reversal that makes laws of nature ontologically prior to individual substances and causally determinative of the natures arising from the motions of their parts is complete in Basso’s natural philosophy. Descartes then takes the additional step of dispensing with the World Soul/ether and rejecting the Aristotelian content of the laws of universal nature in favor of laws of motion developed by Beeckman. I began by highlighting three features of Descartes’s concept of a law of nature that appear to make it both unique and modern in that they set his view apart from his predecessors and anticipate later uses of ‘law of nature’: that Descartes’s three laws of motion are 1) universal, 2) causal, and 3) determinative of all regularities in nature without exception. As seen, Aristotelian predecessors like Piccolomini and Zabarella lay the metaphysical groundwork for 1) and 2) by incorporating Neoplatonic/Stoic commitments to a prior universal nature that manifests itself as an underlying fate/ causal order governing all natural chains of causation. Universal laws, such as that all things perish, play a causal role, though how and when an individual will perish are still causally explained by their particular nature and interactions with surrounding natures. Hence 3) is not a feature of this view. Basso’s reification of the ‘universal nature’ as the World Soul/ether gives him a concrete causal agent which executes the proportions or laws of motion in the divine mind through one universal local motion by means of direct contact with the passive material atoms. Hence Basso’s concept of a law of nature, like Descartes’s, encompasses 1), 2), and 3). We should not be misled by the fact that Basso’s laws of nature remain closer to Aristotelian physics in content

¹⁰¹ Basso (1621, 343–4).

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than the laws of motion Descartes adopts from Beeckman. As seen, at a deeper philosophical level, Basso’s laws of nature differ from Scholastic Aristotelian views for the same reason. Like Descartes, Basso fully rejects the metaphysical order of dependence in Aristotle’s Categories and regards universal laws of nature plus resulting motions of material particles as prior to individual forms and natures. The remaining differences between these two proto-modern views are from a philosophical (albeit not from a scientific or theological) vantage point less significant. They consist in differences in the content of the laws of nature and the fact that whereas for Basso a natural substance (the ether, a very tenuous, active corporeal substance) enacts the laws as God’s instrument, for Descartes, God does so.

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3 Laws of Nature and the Divine Order of Things Descartes and Newton on Truth in Natural Philosophy Mary Domski

1. Introduction During the seventeenth century, God’s existence as an infinite creative being played centrally in how many natural philosophers understood the laws of nature that they were seeking to identify. For mechanical philosophers in particular, who reduced the natural world to matter and motion, God served an especially important role in their characterization of a law-governed world. It was God, as Nancy Cartwright puts it, who ‘inscribed the fundamental laws of mechanics and . . . laid down the initial distribution of matter in the universe.’¹ And it was the natural philosopher who was then charged the task of deciphering the Book of Nature and discovering the fundamental laws that governed the divine order of things. In this respect, as Cartwright emphasizes, ‘God and the Book of Nature were legitimate devices for thinking of laws and relations among them’ during the seventeenth century.² They are also, from our historical vantage point, legitimate devices for understanding the metaphysical underpinnings of early modern laws of nature.³ Of course, the metaphysics only gives us part of the story. The other part of the story has to do with method, and specifically, with identifying methods of inquiry that could allow for the discovery of the fundamental, divinely chosen laws that ¹ Cartwright (1983, 100). ² Cartwright (1983, 101). ³ Evidence that God’s law-governed creation of matter and motion justified the natural philosopher’s search for laws of nature can be found throughout the early modern period. For instance, at the start of the seventeenth century we have Bacon’s discussion in the Novum Organon (1620) of the natural philosopher’s attempt to identify the ‘ideas of the divine’ that have been imprinted onto nature. Along similar lines, at the start of the eighteenth century, Newton claims in the General Scholium (1713) to his Principia mathematica that the order of nature must originate from a divine source.

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govern natural bodies. Here again, God plays a central role, where in this instance, knowledge of God’s essential characteristics, not merely His act of creation, pointed natural philosophers to the types of reasoning and sorts of evidence that could provide insight into the divine order of things.⁴ We see this position forcefully put to use by Roger Cotes in his Editor’s Preface to the second edition of Principia mathematica⁵ as he champions Newton’s ‘experimental,’ empirically grounded method over the ‘rationalist,’ metaphysically grounded approach taken by Descartes in his Principles of Philosophy.⁶ Cotes accepts as given that God is an all-knowing and all-powerful being who created matter and motion, and he also takes it to be evident that the created world ‘could not have arisen except from the perfectly free will of God, who provides and governs all things.’⁷ He then uses this voluntarist reading of God and God’s creation to level a rather damning criticism against Descartes: To pursue Descartes’s ‘rationalist’ methodology, Cotes claims, we either diminish God’s free will or overestimate our human capacity for knowledge. As he puts it, He who is confident that he can truly find the principles of physics, and the laws of things, by relying only on the force of his mind and the internal light of his reason should maintain either that the world has existed from necessity and follows the said laws from the same necessity, or that although the order of nature was constituted by the will of God, nevertheless a creature as small and insignificant as he has a clear understanding of the way things should be.⁸

Unsurprisingly, Cotes maintains that the empirical, ‘experimental’ method that Newton adopts in the Principia is immune from these problems: Reasoning ‘based on phenomena’ as Newton does allegedly preserves both the free choice by which God ordered the created world and our epistemic place below God. As Cotes has it, only by recourse to sensible evidence gathered from the world God actually created— only, that is, ‘by observing and experimenting’—can we identify ‘all the laws that are called laws of nature . . . in which many traces of the highest wisdom and counsel certainly appear, but no traces of necessity.’⁹ Precisely because Newton has pursued this method and relied on ‘causes that truly exist’ as his guide, he has provided in the Principia a ‘solid foundation for his brilliant theories’ and

⁴ Notable examples include Boyle, who emphasizes God’s omnipotence and freedom as he cashes out his program for natural philosophy, and Leibniz, who links God’s rationality with his account of how we ought to investigate natural motions among the phenomena. This is especially apparent in Leibniz’s Discourse on Metaphysics (1686). For more on Boyle’s various statements concerning our knowledge of God and our knowledge of nature, see the very helpful treatment offered by MacIntosh and Anstey (2014). ⁵ Cotes (1713) preface, in Newton (1687/1999). All citations to the Principia mathematica refer to the translation produced by I.B. Cohen and Anne Whitman (1999). ⁶ All citations to the Principles of Philosophy refer to the Miller and Miller translation, which is cited herein as Descartes (1984b). Following standard citation format, I supply page numbers for references to the preface and supply the part and section number for references to the main body of the text. As per Miller and Miller’s convention, squiggly brackets signal text that was added to the 1647 French edition of the Principles. ⁷ Newton (1687/1999, 397). ⁸ In Newton (1687/1999, 397–8). ⁹ In Newton (1687/1999, 397).

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constructed a ‘true philosophy.’¹⁰ In other words, because Newton adopted a method of investigation appropriate to God’s nature, he succeeded in setting forth the ‘laws by which the supreme artificer willed to establish this most beautiful order of the world.’¹¹ Though broad-brushed and clearly biased, Cotes’s polemical remarks raise some important questions about how we are to understand the connection between method and truth in Descartes’s and Newton’s programs for natural philosophy. First, consider Descartes. In line with Cotes’s characterization, Descartes relies on the ‘internal light’ of reason to establish the truth of his three laws of nature. Specifically, and quite famously, in Part II of the Principles of Philosophy, he reasons from God’s existence as the infinite and immutable creator of matter and motion to descriptions of how created bodies must behave.¹² At the same time, Descartes maintains that God’s power is ‘as absolute and free as possible.’¹³ Now, according to Cotes’s indictment, there is an inconsistency here: We cannot simultaneously claim, on the one hand, that God freely willed the creation of nature and, on the other, that we can, by means of human reason alone, discover the laws that govern natural objects. Such a position, Cotes contends, relies on an inaccurate and overinflated account of our rational abilities. And yet, Descartes never denies our epistemic limitations. Indeed, he is quite clear that we cannot know the ultimate purpose, or final causes, of what God has created,¹⁴ and additionally, that we cannot comprehend the precise manner in which God created the natural world.¹⁵ He is equally clear, more generally, that we must always stay mindful that God is infinite and we are finite as we investigate nature.¹⁶ The question, of course, is how exactly Descartes balances his claim to have rationally discovered the true principles of material things with his commitment to the limited capacities of the human intellect. How is it possible, in other words, for Descartes to assert that he has successfully discovered the necessary and law-like way that God conserves material bodies while also maintaining that the human intellect can have only limited insight into God’s absolutely free creation of nature? For Newton, a different question emerges. It is a question in brief of whether pursuing an empirically grounded method of natural philosophy can afford us any absolute or timeless truths at all. In his polemical statements, Cotes maintains that the only means for discovering the ‘laws by which the supreme artificer willed to establish this most beautiful order of the world’ is by relying on sensible and experimental evidence that displays the order of nature that God has freely chosen. However, Newton nowhere claims that the laws he has identified in the Principia correspond to the way God in fact created the world. Indeed, in both the prePrincipia tract De Gravitatione and the General Scholium that is appended to the second edition Principia, Newton suggests rather strongly that any insight into ¹⁰ In Newton (1687/1999, 386, 393). ¹¹ In Newton (1687/1999, 393). ¹² Descartes (1984b, II.36–41). ¹³ I.38. ¹⁴ I.28. ¹⁵ III.45–6.

¹⁶ I.24.

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the divine creation of nature is impossible due to the finite and limited nature of the human intellect and of the human senses.¹⁷ Moreover, in the main text of the Principia, he appears to concede that reasoning based on empirical evidence, and ‘deducing propositions from the phenomena,’ will get us claims that are, at best, provisionally true—claims that are supported by the best evidence we have available and, thus, that are subject to revision and empirical disconfirmation.¹⁸ So, while Cotes emphasizes that Newton has successfully preserved both God’s freedom and our epistemic limitations by pursuing his ‘experimental philosophy,’ the question remains as to whether he has, at the same time, forfeited the robust notion of certain and necessary truth that Descartes’s natural philosophy appears to promise. My aim in what follows is to address these questions by taking a more careful look at the arguments that Descartes and Newton offer in support of their laws of nature. In Section 2, I examine the way that Descartes ‘deduces’ the truth of his laws from our knowledge of God’s perfect nature and pay special attention to how the distinction between understanding and comprehension informs his arguments. Taking this tack we find that, while Descartes clearly takes his laws to be true of the world God created, they are true in an importantly qualified way. Namely, they are not true insofar as they correspond to how God views the world He created. Instead, they capture God’s maintenance of nature so far as we can humanly understand. And as laws that follow from our knowledge of God’s indisputable but also incomprehensible perfections, they are afforded the highest degree of certainty and necessity that we can humanly attain. There is also a qualified sense of truth at play in Newton’s case; however, it is not the sort of contingent truth that we associate with an inductive method. For, on my account, Newton does not argue for his laws by generalizing over a wide store of empirical evidence. Rather, as we see in his presentation of the laws of motion (which will be my focus in Section 3), Newton takes a different tack and offers a more nuanced, two-stage argument: He begins with propositions that he takes to be mathematically and rationally certain, and then relies on empirical evidence to establish that the propositions he has identified accurately describe, i.e., can be applied to, the motions of the sensible world. While not the robust notion of fixed and timeless truth that Descartes claims to offer, the truth of Newton’s laws is a truth characterized by rational certainty and thus closer to Descartes’s notion than we might expect from an ‘empirical’ method. ¹⁷ See Newton (1687/1999, 942) for relevant remarks from the General Scholium and Newton (2004, 27) for remarks from De Gravitatione. ¹⁸ As suggested by the fourth Rule for the Study of Natural Philosophy: ‘In experimental philosophy propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses’ (Newton 1687/1999, 796). In his influential readings of Newton’s Principia, George E. Smith (1999, 2002) has emphasized Newton’s remark that claims are to be taken as ‘very nearly’ [quam proxime] true in natural philosophy. For an illuminating account of how to connect Rule 4 to Newton’s commitment to God’s ordering of nature, see Biener (Forthcoming).

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2. Descartes’s Deduction of the Laws of Nature To establish the true principles of material things in Part II of the Principles, Descartes follows what he describes in Part I as ‘the best method of philosophizing’ and ‘deduce[s] the explanation of things created by [God] from the knowledge of God Himself.’¹⁹ The knowledge of God that is fundamental to the arguments of Part II is the knowledge that, as a perfect being, God is ‘not only immutable in His nature, but also immutable and completely constant in the way He acts.’²⁰ Initially, Descartes combines this knowledge of God’s constant and unchanging activity with what is known to be true of matter and motion, and reasons that the amount of motion in the universe always remains constant. As he presents it, the truth of this conservation principle is entailed by our knowledge that [1] God is the primary cause of all movement in the world; [2] God created matter with movement and rest; [3] God conserves, or ‘normally participates’ in, nature; and finally, [4] as a supremely perfect being, God is immutable and completely constant in the way He conserves nature.²¹ From these truths concerning God and His creation, ‘it follows,’ Descartes asserts, ‘that it is completely consistent with reason’ that God ‘always maintains in [matter] an equal quantity of motion.’²² Immediately following this argument, Descartes reports that ‘from this same immutability of God, we can obtain knowledge of the rules or laws of nature, which are the secondary and particular causes of the diverse movements that we notice in individual bodies.’²³ And thus, in his proofs for the three laws of nature, he pursues the same general argument strategy that was used to establish the conversation of the quantity of motion: In each case, God’s constant and unchanging nature sets the standard for what is reasonable to think about natural motion. Take, for instance, the first law of nature, according to which ‘each thing, provided that it is simple and undivided, always remains in the same state as far as is in its power, and never changes except by external causes.’²⁴ According to the proof for this law, we ‘are easily convinced’ that any change to a body, whether to its external shape or to its state of being at motion or at rest, must be the result of some external cause. For, in the absence of any such cause, a body will remain in its ‘natural’ state, as that which remains unchanged, because it is conserved by a God who is ‘immutable and completely constant in the way He acts.’

¹⁹ I.24. ²⁰ II.36. ²¹ II.36. ²² II.36. ²³ II.37. For more on how we might understand Descartes’s distinction between God as the primary cause of motion and the laws as the secondary cause of motion, see especially Hattab (2000) as well as Ott (2009) and Schmaltz (2008). ²⁴ II.37. Descartes first presents his laws of nature in the earlier Le Monde (ca. 1633), a text in which he promotes a Copernican sun-centered model of the universe, and which he abandoned after hearing of Galileo’s condemnation by the Catholic Church. In Le Monde, the laws are listed in a different order and, in some cases, as I indicate below, there are notable differences between what is claimed in Le Monde and what is argued in the Principles. For more on these differences, see Gaukroger’s introduction to Descartes (1998).

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In a similar vein, the truth of God’s unchanging maintenance of nature supports our acceptance of the second law, according to which ‘all movement is, of itself, along straight lines; and consequently, bodies moving in a circle, always tend to move away from the center of the circle which they are describing.’²⁵ In the argument for this law, Descartes makes explicit appeal to the ‘proper’ account of motion that had been established in II.25, and emphasizes that movement does not happen in an instant. Motion involves the transfer of a body from the vicinity of one collection of contiguous and resting bodies to the vicinity of another, and this can only occur over time.²⁶ It is also the case, Descartes tells us, that, at every instant of a moving body’s path, there is in the body an inclination to move in some direction, an inclination that is linked to ‘the immutability and simplicity of the operation by which God maintains the movement in matter.’²⁷ In particular, since God maintains movement ‘precisely as it is at the very moment at which He is maintaining it, and not as it may perhaps have been at some earlier time,’²⁸ the instantaneous inclinations that God provides must be directed in a straight line.²⁹ Thus, putting the pieces together, ‘it is obvious’—or it should be obvious—that ‘each part of matter, considered individually, tends to continue its movement only along straight lines, and never along curved ones.’ Finally, God’s immutability is a key element in the proof for the second part of the third law of nature, according to which a moving stronger body that comes into contact with a weaker one loses as much motion as it transfers to the weaker body.³⁰ As Descartes presents it, the truth of this principle stems from the truth of the second

²⁵ II.39. Descartes restates the second law as follows: ‘that each part of matter, considered individually, tends to continue its movement only along straight lines, and never along curved ones; even though many of these parts are frequently forced to move aside because they encounter others in their path, and even though, as stated before [cf. II.33], in any movement, a circle of matter which moves together is always in some way forced’ (II.39). ²⁶ According to II.25, ‘movement, according to the truth of the matter rather than in accordance with common usage’ is ‘the transference of one part of matter or of one body, from the vicinity of those bodies immediately contiguous to it and considered as at rest, into the vicinity of [some] others.’ ²⁷ II.39. ²⁸ II.39. ²⁹ The second law of the Principles is presented as the third rule of nature in Le Monde, where Descartes is more explicit about the link between God’s activity and a body’s rectilinear inclination to motion. He claims, in particular, that God acts simply in His conservation of motion insofar as He conserves motion at each instant through simple, straight-line motions (Descartes 1998, 29–30). ³⁰ Descartes makes no explicit appeal to God’s immutability in establishing the first part of this law, viz., that a moving weaker body coming into contact with a stronger one will lose none of its motion. Instead, he relies on the rational distinction between motion and the direction of motion that was alluded to in the argument for the second law, and he takes it to be evident that, upon coming into contact with a stronger body, a weaker body will change its direction, but not its quantity of motion (II.41). In Le Monde, Descartes does not differentiate between the cases of a weaker body striking a stronger one and a stronger one striking a weaker one. In both sorts of collision, Descartes maintains that there is an equal gain and loss of motion in the two bodies: ‘when one of these [moving] bodies pushes another it cannot give the other any motion except by losing as much of its own motion at the same time; nor can it take away any of the other’s motion unless its own is increased by the same amount’ (Descartes 1998, 29).

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law, the truth that the universe is a plenum, i.e., that there is no vacuum in nature,³¹ and the truth that God ‘always uninterruptedly maintain[s] the world by the same action by which He created it.’³² Combining these claims with the implicit assumption that a stronger body loses at least some of its motion when striking a weaker one, it should be manifest, according to Descartes, that the stronger body loses as much motion as it transfers to the weaker one. And thus, with this law, Descartes has established that God’s immutable maintenance of nature guarantees that the quantity of motion is conserved, not only in the universe in general, but also in local collisions of this sort. As is clear from the arguments sketched above, Descartes ‘deduces’ the necessary and law-like way that God maintains the motion of the material bodies that He has created through rational consideration of God’s existence as an immutable being.³³ More generally, the arguments for each of the three laws direct us to what is ‘obvious’ and ‘consistent with reason’ given the metaphysical and physical truths that had been established in earlier portions of the Principles.³⁴ Recall now the worry that Cotes voices about Descartes’s methodology, and specifically, his attempt to discern how the divinely ordered world operates by ‘relying only on the force of his mind and the internal light of his reason.’ The problem, from Cotes’s voluntarist perspective, is that for this sort of ‘rationalist’ methodology to succeed requires that we assume that human reason has the capacity for insight into the divine order of things. It is to assume, in other words, that human reason, unto itself and without recourse to empirical evidence of how God actually created the world, is equipped to discern a ‘clear understanding of the way things should be.’ Such an assumption, Cotes alleges, is inconsistent with our human status as ‘small and insignificant’ creatures. Before engaging Cotes’s central worry, it’s important to note that, in a significant respect, he is forwarding an uncharitable characterization of Descartes’s methodology. For Cotes’s suggestion to the contrary, Descartes nowhere asserts that human reason can unto itself establish the true principles of material bodies. Instead, he consistently maintains that our rational capacity for discovering the truth is given to us by God and, moreover, he claims that our rational ability to attain knowledge requires that we rely on clear and distinct innate ideas, which themselves originate from a divine source.³⁵ As such, any knowledge attained by the finite human intellect is thoroughly dependent on an infinite God, and in this respect at least, Descartes’s position is thoroughly consistent with our ‘small and insignificant’ status relative to God. ³¹ II.16. ³² II.42. ³³ For more on the ‘deductive’ ideal that guides Descartes’s natural philosophy, see McMullin (2008), who emphasizes Descartes’s resistance to making explicit appeals to empirical evidence as he establishes his laws of nature. ³⁴ In this respect, the text follows the step-wise procedure famously described by Descartes in the preface to the 1647 French edition of the Principles: ‘Philosophy as a whole is like a tree; of which the roots are Metaphysics, the trunk is Physics, and the branches emerging from this trunk are all the other branches of knowledge’ (Descartes 1984b, xxiv). ³⁵ Cf. I.24, I.28, and I.30.

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Nonetheless, even with this qualification in place, there is a key element of Cotes’s critique that lingers: If Descartes is maintaining that human reason is able to discern laws that capture the divine order of things, then even if we must in some way depend on God for this knowledge, he remains committed to the apparently problematic claim that, under the proper circumstances, our limited and finite rational abilities are on equal epistemic footing as the divine intellect—that if we reason in the correct way, we can gain insight into God’s view of creation and reach, as Cotes puts it, ‘a clear understanding of the way things should be.’ The arguments for the three laws of nature seem to bear this out. In each case, Descartes reasons from God’s nature to what must be concluded about God’s maintenance of matter and motion, e.g., that He is responsible for the rectilinear inclinations to motion at each instant of a body’s path, and that He conserves the quantity of motion, both in the universe and in specific sorts of local collisions. The laws thus specify how a supremely perfect God conserves, or ‘normally participates’ in, nature, and even if attaining this knowledge of God’s activity requires divine assistance, it seems, as per Cotes’s critique, that Descartes’s God has provided our finite and dependent intellects the resources to achieve the same knowledge that ought to be reserved for the infinite and independent intellect of God. Here, I want to suggest, Cotes’s critique misses a crucial element of Descartes’s program, and it does, because the truth that Descartes associates with his laws of nature is not the truth that Cotes alleges. Specifically, Descartes is not claiming that his laws communicate how the world is perceived by the infinite mind of God. Indeed, with Descartes’s admission that we cannot know how, or to what end, God originally created matter and motion, the truth that Descartes associates with the laws of nature must be of a different variety. The laws are certain and necessary, to be sure, but they are certain and necessary from our epistemic vantage point. In other words, and in line with the distinction between ‘understanding’ and ‘comprehension’ that informs Descartes’s methodology, the laws are meant to capture how a limited, dependent, and finite intellect must understand God’s maintenance of matter and motion.³⁶ As we saw above, our knowledge of the three laws of nature is dependent on our knowledge that, as a perfect being, God is ‘not only immutable in His nature, but also immutable and completely constant in the way He acts.’³⁷ Crucially, this knowledge of God’s perfect nature has a unique character: Stemming from our reflection on the innate idea of God, it is knowledge that is the most clear and most distinct that we can humanly attain, and yet, it is knowledge that does not provide us comprehension ³⁶ The reading I offer here relies on there being a difference between what we must necessarily conclude about God’s activity, and any necessity that might characterize God’s creative action, i.e., on what it is or isn’t metaphysically possible for God to create. As should already be clear, my focus is on what it is we must conclude when we reason from God’s nature to the bodies He created. For further discussion of the metaphysical issue, see Schmaltz (2014). ³⁷ II.36.

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of God’s perfections. We reflect on this idea and know, for instance, that God is ‘eternal, omniscient, omnipotent, the source of all goodness and truth, the creator of all things, and . . . that He has in Him all those things in which we can clearly observe some perfection which is infinite or limited by no imperfection.’³⁸ This understanding of God’s nature, while indubitable and certain, remains knowledge that we cannot grasp. For instance, Descartes maintains that we cannot comprehend God’s absolute infinity, because as finite creatures, our comprehension reaches only as far as potential infinity—to processes or quantities that continue without end.³⁹ But still, we understand that God is absolutely infinite, because this perfection is revealed to reason. In the terms of the Third Meditation, God’s actual and not merely potential infinity is among the divine perfections ‘which I cannot grasp, but can somehow reach in my thought.’⁴⁰ And, as Descartes tells us in the Principles, so it is with all of God’s perfections: That even though we do not comprehend the nature of God, nevertheless, His perfections are known to us more clearly than any other thing. And this is sufficiently certain and evident to those who have been accustomed to contemplating the idea of God and to noticing His supreme perfections. For although we do not comprehend these perfections, because of course it is of the nature of the infinite not to be comprehended by us who are finite, we can however understand them more clearly and more distinctly than any corporeal things; because they fulfil [sic] our mind more, and are more simple, and are not obscured by any limitations.⁴¹

Lacking comprehension, of course, does not provide us reason for doubting what is presented to the intellect. In line with the clear and distinct perception rule, it is our inability to doubt, not our inability to comprehend, that signals what is objectively true of a world that exists independently from and outside of the perceiving, judging mind.⁴² And in the case of the idea of God, our knowledge of His supremely perfect nature is manifest, obvious, and evident to reason. It is a certainty that we must ³⁸ I.22. ³⁹ See especially the discussion in I.26–7 of the Principles, where Descartes urges caution when we come across natural, created things (such as the number of stars and the extension of space) that appear to us to go on without end. In these cases, since we are reasoning about what we comprehend by means of our finite intellects, we should use the term ‘indefinite.’ Only when we reason about God, whose nature is revealed to the intellect, should we use the term ‘infinite.’ ⁴⁰ AT VII, 52; CSM II, 35. See also the Appendix to the Fifth Replies, where Descartes states explicitly that the idea of God can be perceived but cannot be grasped: ‘Since the word “grasp” implies some limitation, a finite mind cannot grasp God, who is infinite. But that does not prevent him having a perception of God, just as one can touch a mountain without being able to put one’s arms round it’ (AT IX A210; CSM II, 273–4). As I read Descartes, having this ‘perception’ of the idea of God is both sufficient and necessary for ‘understanding,’ i.e., knowing to be true, that God has the perfections that are associated with the idea of God. ⁴¹ I.19. ⁴² The clear and distinct perception rule is offered in I.30 of the Principles. Having established in I.29 that God is not the cause of errors, Descartes writes: ‘I.30. That it follows from this that all things which we clearly and distinctly perceive are true, and that the doubts previously listed are removed. And from this it follows that the natural enlightenment or the faculty of knowing given to us by God, can never attain any object which is not true, insofar as it is clearly and distinctly perceived.’

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accept as true, and we must accept it because our God-given ‘natural enlightenment,’ not our human standards for comprehension, indicates what counts as true and genuine knowledge.⁴³ This account of ‘understanding’ helps explain why Descartes takes our incomprehensible and indubitable knowledge of God to be foundational in his ‘deductions’ of the laws of nature—and why, more generally, he declares that ‘we shall be following the best method of philosophizing if we strive to deduce the explanation of things created by Him from the knowledge of God Himself.’⁴⁴ Not only is it the case that the procedure Descartes describes maps onto the metaphysical order of nature, proceeding from God as the first cause of matter and motion to His willfully created effects. From an epistemic standpoint, to proceed from our knowledge of the Creator to the bodies He has created, we are positioned to make claims about the order of nature that are as indubitable and certain as our understanding of God’s nature. Of course, the same qualification about our knowledge of God will be carried over to the knowledge of bodies that we ‘deduce’ from God’s nature: Our understanding of God’s perfections will lead us to an understanding, not a comprehension, of the way natural objects behave. Grasping what is the case cannot be achieved here, because as in the case of knowing God, we are not wholly relying on the resources of the finite intellect as we identify the principles of material things. We are relying on what is revealed to be true through consideration of God’s perfections.⁴⁵ Thus, just as we know that God is actually infinite without comprehending God’s actual infinity, in the same vein, we know that objects created by God must behave in a particular way (namely, as described by the three laws of nature) even though we do not grasp these truths. We do not, for instance, comprehend the means by which an infinite and immaterial God can interact with finite and material bodies, and yet we know from our limited rational standpoint that there is such an interaction.⁴⁶ And, as Descartes

⁴³ Cf. I.26–8. ⁴⁴ I.24. ⁴⁵ I don’t deny that it is in principle possible to carry out a deduction that proceeds from a claim that is both true and incomprehensible to one that is true and comprehensible in Descartes’s sense. For instance, geometrical proofs might involve axiomatic claims, which are to be accepted as true without proof, and bring us to conclusions that are constructible and thus graspable. The proof that the angles of a triangle sum to two right angles would be such a case. The point I emphasize here is that by the standards that Descartes adopts, we cannot reach a truth that is ‘graspable’ when we reason from God’s incomprehensible nature, because on his account, this reasoning relies solely on what is revealed and understood from our innate ideas. As such, when we proceed from knowledge of God’s perfections to knowledge of how God maintains the material bodies He has created, we are not to hold the conclusions we reach to the standards of comprehension. ⁴⁶ In his letters to Princess Elisabeth from 21 May 1643 and 28 June 1643, Descartes indicates that human reason is not equipped with the conceptual apparatus, or ‘primitive notions,’ to comprehend how an immaterial thing can interact with a material one—in this specific case, how an immaterial soul can interact with a material body. Of course, we cannot deny that there is such an interaction; however, according to Descartes, our attempts to explain this interaction continue to be inadequate, because the explanations rely on notions that ought to be associated with the body alone. For Elisabeth’s letters and Descartes’s replies, see Shapiro (2007, 61–71).

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maintains, we know this to be as certain as any truth that is revealed to us through our divinely supplied clear and distinct ideas: That all things which have been revealed by God must be believed, although they may surpass our power of comprehension. Thus, if it happens that God reveals to us something, concerning Himself or other things, which exceeds the power of our understanding (as the mysteries of the Incarnation and the Trinity already do); we shall not refuse to believe those things, although we {perhaps} do not clearly understand them. And we shall not wonder in the slightest that there are many things, both in His boundless nature and also in the things created by Him, which surpass our power of comprehension.⁴⁷

Recall that the force of Cotes’s critique centered on the alleged incompatibility of asserting rationally gained human knowledge of nature while also maintaining our ‘small and insignificant’ place in nature. It is a legitimate worry in general. However, it is not one that can be fairly applied to Descartes’s project, because he embeds our epistemic position below God in the very notion of ‘understanding’ on which his rational method for discerning truth relies. As a consequence, he identifies certain and indubitable laws that do not promise the complete and perfect insight into natural bodies and motion that are reserved for God’s infinite intellect. Instead, the three laws of nature he identifies are true for us—for rational beings who are equipped with a Godgiven capacity for ‘natural enlightenment’ and who have access to divinely supplied innate ideas. In this respect, Descartes’s laws of nature are true of the natural world that God freely created, but what the laws communicate is to be understood in an importantly qualified sense: They are accurate, fixed, and timelessly valid descriptions of how matter and motion behave, so far as human reason is equipped to understand.

3. Newton’s Deduction of the Laws of Motion Even with this qualification in place, Descartes’s ‘rationalist’ methodology allows him to associate a robust notion of truth with his laws of nature. Following his argument strategy, the laws serve as unrevisable and timeless descriptions of the divine order of things, because they are ‘deduced’ from the timeless, immutable, and perfect knowledge of God’s nature. The question as we now turn to Newton’s laws of nature is whether his empirical method of ‘deducing propositions from the phenomena’ can yield laws that are true in the same sense.⁴⁸ To be sure, and as per Cotes’s assessment, Newton has grounds for claiming that his principles capture the behaviors of bodies that populate the sensible world that God created since he reasons to his laws from ⁴⁷ I.25. ⁴⁸ In the General Scholium to the Principia, Newton uses the slogan ‘deduction from the phenomena’ to characterize the method he uses to establish the laws of motion as well as the law of gravity and the properties of bodies (Newton 1687/1999, 943). In what follows, I narrow my focus on the laws of motion to bring out the similarities and differences between Newton’s and Descartes’s strategies for establishing true laws of nature. In Domski (Forthcoming), I offer a more general treatment of ‘deductions from the phenomena’ and connect Newton’s argument for the laws of motion with his arguments for universal gravitation and the properties of bodies.

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sensible evidence—from what we witness in nature and gather from experiments. At the same time, having an empirical foundation for his natural philosophy appears to prevent Newton from claiming that his principles capture what is fixed and timeless in nature, because as laws based on the current evidence available, they would be subject to revision should further evidence demand. Consequently, the best we can say, it seems, is that Newton’s laws are provisionally true descriptions of the divine order of things, and that their truth is contingent, not merely on our human rational or sensory abilities, but on sensory and experimental information that itself, and by its very nature, is subject to elaboration and modification. The apparently provisional character of Newton’s laws has given rise to longstanding interpretative questions about the status of his three laws of motion. These laws are presented at the opening of the Principia, immediately following the Definitions section, and being included in a section entitled ‘Axioms, or Laws of Motion,’ the laws are presented as foundational to Newton’s program of natural philosophy. Namely, the laws serve as descriptions of the fundamental properties of motion—inertia, the proportionality of force and mass, and the equality of action and reaction—that apply to all material bodies, and as ‘axiomatic,’ they are descriptions of motion that must be accepted in order for the rest of the work to succeed. However, from a methodological standpoint, there is a question of why Newton takes the laws to be axiomatic, and more pointedly, whether he offers good reason to accept them as principles that apply to all material bodies, universally and without exception. The first and third laws of motion help clarify what’s at stake here, because in presenting these laws, Newton makes explicit reference to empirical cases and appears to be offering empirical justification for their acceptance. For instance, after stating the law of inertia, Newton mentions moving projectiles, spinning tops, and planetary bodies as instances where motion is impeded by impressed forces: Law 1: Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. Projectiles persevere in their motions, except insofar as they are retarded by the resistance of the air and are impelled downward by the force of gravity. A spinning top [trochus], which has parts that by their cohesion continually draw one another back from rectilinear motions, does not cease to rotate, except insofar as it is retarded by the air. And larger bodies—planets and comets—preserve for a longer time both their progressive and their circular motions, which take place in spaces having less resistance.⁴⁹

In a similar vein, Newton references a finger pressing a stone, a horse-drawn rope, and two colliding bodies when presenting the action-reaction law: Law 3: To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction. Whatever presses or draws something else is pressed and drawn just as much by it. If ⁴⁹ Newton (1687/1999, 416).

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anyone presses a stone with a finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse will (so to speak) also be drawn back equally toward the stone, for the rope, stretched out at both ends, will urge the horse toward the stone and the stone toward the horse by one and the same endeavor to go slack and will impede the forward motion of the one as much as it promotes the forward motion of the other. If some body impinging upon another body changes the motion of that body in any way by its own force, then, by the force of the other body (because of the equality of their mutual pressure), it also will in turn undergo the same change in its own motion in the opposite direction.⁵⁰

One might reasonably conjecture, as Alan Shapiro has, that these passages indicate that Newton has ‘deduced’ the laws of motion ‘directly from the phenomena.’⁵¹ On this interpretation, the empirical and observable evidence Newton presents in the passages above is meant to convince us that the laws should be taken as true—that, to paraphrase Descartes, Newton is arguing that it is completely consistent with sensory evidence to accept that the laws are accurate descriptions of how all physical bodies behave. Newton is thus to be read as making a standard inductive move and offering a two-step argument: He first gathers evidence that bodies we commonly observe act according to the laws, and then, from this evidence he generalizes the truth of the laws to all observable bodies. It is from this generalizing step that the laws earn their axiomatic status insofar as Newton is stipulating that the laws are generally true, even though we might, at a future time, encounter an empirical exception to the behaviors that the laws describe. There are some clear advantages to such a reading. It is consistent with the directive of Rule 4 that, in ‘experimental philosophy,’ propositions should be ‘considered either exactly or very nearly true notwithstanding any contrary hypotheses.’⁵² Moreover, as Shapiro urges, this reading fits well with Newton’s project to establish a program of natural philosophy that could replace ‘hypothetical’ philosophies that rely on imaginary causes.⁵³ All the same, this reading leaves us with questions about the axiomatic status of the laws, because if Newton is offering an inductive argument to support their acceptance, they would, at best, be provisionally axiomatic, that is, they could be accepted as true only for so long as there is no empirical evidence to the contrary. As a result, the entire program of natural philosophy pursued in the Principia would be as provisional as the axiomatic laws on which it is based.

⁵⁰ Newton (1687/1999, 417). ⁵¹ Shapiro (2004, 212; emphasis added). ⁵² Newton (1687/1999, 796). ⁵³ Newton explicitly distinguishes the laws of motion from ‘hypotheses’ in a draft letter from March 1713: ‘these laws in being deduced from phenomena by induction and backed with reason and the three general rules of philosophizing are distinguished from hypotheses and considered as axioms’ (Newton 2004, 119–20).

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I want to suggest a more nuanced way to read Newton’s use of empirical evidence in his presentation of the laws of motion, one that allows us to better understand why he calls the laws ‘axioms’ and also to better appreciate the affinity between Newton’s methodology and Descartes’s. Rather than read Newton as offering a two-step argument that proceeds from evidence to generalization, there is textual evidence for reading him as offering a two-step argument that proceeds from certainty to application. Specifically, we can take the laws to be axiomatic insofar as they are rationally certain, and we can take them to be true insofar as evidence shows us that the laws apply to behaviors in the natural world. On this interpretation, what grants the laws their axiomatic status is our rational certainty—our inability to doubt—that, by the standards of mathematics, they accurately describe the motion of bodies. The truth of the laws is then ‘deduced from the phenomena’ insofar as observable evidence shows that they capture the behaviors we witness in the empirical world.⁵⁴ That Newton associates a mathematical standard of certainty with the laws of motion is strongly suggested by his claim, at the opening of Book 3, that, in addition to the propositions of Books 1 and 2, the laws of motion are among the ‘strictly mathematical’ principles upon ‘which the study of philosophy can be based.’⁵⁵ To unpack what Newton means by ‘strictly mathematical,’ we can turn to his various characterizations of the ‘mathematical’ approach to motion that he is forwarding in the opening portions of the Principia.⁵⁶ In these discussions, Newton consistently contrasts his mathematical approach with a ‘physical’ one and, he tells us, from a mathematical point of view, we are to consider only the ‘quantities and mathematical proportions’ of motions and forces, not ‘the species of forces and their physical

⁵⁴ I take ‘phenomena’ to include empirical evidence in general, whether it’s evidence gathered from everyday observations of the world or from carefully crafted experiments. This reading is consistent with the sort of evidence to which Newton refers in his discussion of the laws of motion. William Harper (2011) relies on a much more technical reading of ‘phenomena’ in his provocative reading of Newton’s ‘deduction from the phenomena,’ because his primary goal is to explain the role of the Phenomena section in Newton’s ‘deduction’ of universal gravitation. According to Harper, the six phenomena in that section should be read, not as mere observable data, but as ‘patterns exhibited in open-ended bodies of data,’ i.e., as data that has been organized by Newton in a particular way (Harper 2011, 23). ⁵⁵ Newton (1687/1999, 793). ⁵⁶ Recent commentators, such as Steffen Ducheyne (2012) and Andrew Janiak (2008), have drawn on Newton’s distinction between ‘the mathematical’ and ‘the physical’ in their interpretations of the method Newton uses in the Principia to establish universal gravitation. Ducheyne also provides a reading of the laws of motion that is meant to explain how they fit into the mathematical project of the opening books of the Principia. On his reading, the laws of motion are ‘theoretical principles . . . that have empirical support’ from experimental testing, and they are to be considered abstract principles that are legitimately a part of the mathematical project Newton adopts in Book 1. While I don’t dispute the general point he puts forward, I raise worries about Ducheyne’s reading of the laws as both abstract and empirically supported in Domski (2013, 328–30). On the reading I forward here, I aim to give a fuller explanation of the connection between the mathematical status of the laws and the empirical evidence Newton offers in support of them.

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qualities.’⁵⁷ The point is also emphasized in the final paragraph of Definition 8, where Newton again contrasts the ‘mathematical point of view’ with the investigation of physical forces: I use interchangeably and indiscriminately words signifying attraction, impulse, or any sort of propensity toward a center, considering these forces not from a physical but only from a mathematical point of view. Therefore, let the reader beware of thinking that by words of this kind I am anywhere defining a species or mode of action or a physical cause or reason, or that I am attributing forces in a true and physical sense to centers (which are mathematical points) if I happen to say that centers attract or that centers have forces.⁵⁸

From the ‘mathematical point of view,’ ‘body’ and ‘force’ signify something different than when these notions are used in a ‘physical sense.’ In the case of the latter, ‘body’ is a physical substance with a variety of sensible properties, and the action of such bodies is brought about by some ‘physical cause or reason.’ For instance, we see a billiard ball move, and we take this motion to be the effect of some physical cause, whether the collision with another ball or the striking of a cue stick. But this is not how Newton is using ‘body’ in the opening portions of the Principia. He is treating ‘body’ in a mathematical sense, as a spatially located object that has some quantity of matter, some quantity of motion, and some forces attributed to it. In other words, the ‘bodies’ to which Newton refers in the ‘Axioms, or Laws of Motion,’ and also in Books 1 and 2 of the Principia, are ones that have those properties specified by the Definitions.⁵⁹ And whatever quantities of matter, motion, and force these might be, a mathematical body does not have any real, physical forces acting on it, because it is not a physical body. As mathematical, it is an idealized version of a real body, one that shares only a selection of quantifiable properties with physical objects, and one that, as Newton puts it, is attributed forces and motions that ‘follow from any conditions that may be supposed.’⁶⁰ ⁵⁷ Newton (1687/1999, 588). The passage reads in full: ‘I use the word “attraction” here in a general sense for any endeavor whatever of bodies to approach one another, whether that endeavor occurs as a result of the action of the bodies either drawn toward one another or acting on one another by means of spirits emitted or whether it arises from the action of aether or of air or of any medium whatsoever—whether corporeal or incorporeal—in any way impelling toward one another the bodies floating therein. I use the word “impulse” in the same general sense, considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.’ ⁵⁸ Newton (1687/1999, 408). ⁵⁹ Newton presents definitions of eight terms: [1] the quantity of matter, [2] the quantity of motion, [3] inherent force, [4] impressed force, [5] centripetal force, [6] the absolute quantity of centripetal force, [7] the accelerative quantity of centripetal force, and [8] the motive quantity of centripetal force (cf. Newton 1687/1999, 403–8). In this section, Newton also emphasizes that he is taking a mathematical approach to forces and motions. For instance, in the discussion of Definition 8, he elaborates on the differences between motive, accelerative, and absolute centripetal forces, and in reference to the latter, he explains, ‘This concept [of absolute force] is purely mathematical, for I am not now considering the physical causes and sites of forces’ (Newton (1687/1999, 407). ⁶⁰ Newton (1687/1999, 588). The process of abstraction by which we reach this ideal, mathematical version of body is mentioned in the opening sections of De Gravitatione, which I have treated in more detail in Domski (2012). The only explicit reference to abstraction in the Principia that I’m aware of comes

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We can now get a better handle on what Newton means when he says that the laws of motion are ‘strictly mathematical’ principles upon ‘which the study of philosophy can be based.’⁶¹ Consistent with the remarks above, they are ‘strictly mathematical’ insofar as they are laws that describe the behavior of mathematical bodies, viz., idealized bodies that have quantifiable properties that can be reasoned about without consideration of what physical causes may give rise to these properties. And as bodies considered in abstraction from any physical circumstances, the laws that describe their behavior are evident and certain no matter the empirical evidence available. Put differently, as mathematically evident and mathematically certain laws of motion, they require no empirical justification and stand as immune from empirical disconfirmation, because, ultimately, they are not laws about empirical bodies at all. They are ‘strictly mathematical’ principles concerning mathematical bodies that, by the standards of mathematical reasoning, are beyond rational doubt.⁶² This, of course, raises questions about the empirical evidence that Newton references in the presentations of the first and third laws. Such evidence would appear entirely unnecessary in the mathematical context of the opening sections of the Principia: The behavior of spinning tops, colliding bodies, and the like do not make Newton’s mathematically axiomatic claims about motion any more evident or more certain. While perhaps tempting, then, to read these as heuristic examples meant to clarify the meaning of the laws to the reader, we see Newton assigning empirical evidence a more substantial role in his remarks from the Scholium to the ‘Axioms, or Laws of Nature.’ In this context, Newton aligns experimental results with ‘truth,’ which suggests, I claim, that the phenomena presented in the discussion of the laws of motion are meant to show that these mathematically certain and axiomatic laws are true of physical objects—that although the laws are not about physical bodies, they are laws that can be applied to real bodies that are being acted on by physical causes. To clarify, consider Newton’s discussion of the rules of impacts and collisions in the Scholium. He claims that these rules describing ‘the collisions and reflections of hard bodies’ have been mathematically demonstrated from the laws of motion and their corollaries by ‘Sir Christopher Wren, Dr. John Wallis, and Mr. Christiaan in the Scholium to the Definitions. Here, as Newton discusses the differences between absolute and relative places and motions, he remarks that, in natural philosophy, ‘abstraction from the senses is required’ (Newton 1687/1999, 411). ⁶¹ Newton (1687/1999, 793). ⁶² Another way to make the point is that, for Newton, the laws of motion could not be otherwise for the mathematical bodies he has defined. These laws are the ones that obtain evidently and without need of proof. To be sure, we could imagine a system of rational mechanics that adopts different foundational and ‘axiomatic’ laws of motion. However, my point here is that any such system would have to define motions, forces, and bodies differently than Newton does. My thanks to Zvi Biener for urging me to clarify this point. For the role of certainty in Newton’s mathematical practice, see the trenchant account in Guicciardini (2009).

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Huygens, easily the foremost geometers of the previous generation’ (424–5).⁶³ He then continues: Wallis was indeed the first to publish what had been found, followed by Wren and Huygens. But Wren additionally proved the truth of these rules before the Royal Society by means of an experiment with pendulums, which the eminent Mariotte soon after thought worthy to be made the subject of a whole book.⁶⁴

Notice that there are two stages of reasoning suggested here. First, we have ‘the foremost geometers of the previous generation’—Wren, Wallis, and Huygens— mathematically demonstrating the rules of collisions and reflections. Then, with Wren’s experiment, we have an empirical stage that proves ‘the truth of these rules.’ What’s suggested here is not that the experimental evidence supports or contributes to the mathematical certainty of what’s been demonstrated. It’s that Wren’s experimental results prove that the mathematical theory can be applied to the observed motion of pendulums. That is, Wren proved by experiment that the mathematically demonstrated rules accurately describe the behavior of real, existing bodies. Throughout the Scholium, Newton continues to link the notion of ‘proof ’ and also the notion of ‘testing’ with experimental results. There is discussion of experiments ‘designed to prove’ the rules of collision, of testing the rules of impact by means of ‘tightly wound balls of wool strongly compressed,’ and also of testing what follows from the first law of motion ‘with a lodestone and iron.’⁶⁵ In these instances, as above, Newton’s remarks suggest that what experimental, observable evidence provides is confirmation that mathematically demonstrated propositions concerning motion can be applied to physical objects. To be sure, these are not everyday physical objects. In our experimental set up, Newton urges that we take care to diminish the effects of physical causes, such as air resistance, on the motion of the bodies that we are observing so that the experiments can provide ‘proof ’ that actually existing bodies will behave as the mathematical theory describes.⁶⁶ As such, the experiments are meant to test whether what is mathematically certain can be applied to the actual, albeit idealized, state of affairs. My suggestion is that Newton is urging the same point when he refers to empirical cases in the presentation of the first and third laws: He is offering observable evidence that ‘proves the truth’ of the axiomatic, mathematically certain laws—evidence that ⁶³ Newton (1687/1999, 424–5). ⁶⁴ Newton (1687/1999, 424–5; emphasis added). The rules of collision proven by Wallis and Wren were published in the Philosophical Transactions of the Royal Society in 1668. Huygens’s demonstration of the rules was published first in the Journal des scavans in 1668 and then in Philosophical Transactions in early 1669. For an illuminating comparison of their demonstrations, see Bertoloni Meli (2006, 227–40). ⁶⁵ Cf. Newton (1687/1999, 427–8). ⁶⁶ See, for instance, Newton’s discussion of how to craft pendulum experiments to ‘prove’ the rules of collision and impact. He offers a series of steps that can be taken to establish experiments that account for empirical factors, such as air resistance and elastic force. Such steps, he claims, ‘make it possible to make all our experiments, just as if we were in a vacuum’ (Newton 1687/1999, 425).

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shows that the behaviors the laws describe are instantiated among actually existing bodies. In this case, of course, the examples are not drawn from carefully crafted experiments that account for interfering empirical factors. We have examples involving moving projectiles, spinning tops, laboring horses, and fingers pressing stones. These are phenomena that can be seen in the course of our everyday experience and that involve really existing physical objects that are being acted upon by real and physical forces. Consequently, unlike the experimental proofs and tests that are presented in the Scholium, these everyday empirical instances don’t simply indicate that the mathematical theory governs bodies with select mathematical properties. More significantly, they indicate that the axiomatic laws of motion govern the behavior of actually existing, non-idealized natural bodies. In this respect, Newton is showing that these ‘strictly mathematical’ laws of motion are laws that capture what we encounter in the course of even our most basic human experience, and thus, that merit being axiomatic for a program of natural philosophy aimed at explaining all the motions and forces of nature. To be sure, the notion of truth at play here is not the timeless and universal truth that Descartes deploys. It is truth as applicability, and showing that the laws of motion are true of, i.e., apply to, some empirical cases is clearly not sufficient to establish that they apply to all empirical cases. This is where the inductive step enters for Newton. He ‘deduces’ the truth, or application, of the laws of motion ‘from the phenomena,’ and then, as per remarks from a 1713 draft letter to Cotes, these principles ‘are made general by induction’: ‘Experimental philosophy argues only from phenomena, draws general conclusions from the consent of the phenomena, and looks upon the conclusion as general when the consent is general without exception, though the generality cannot be demonstrated a priori.’⁶⁷ As Newton indicates here, knowing that the laws of motion hold of any bodies requires the ‘consent of the phenomena,’ i.e., it requires empirical confirmation. To go further and say that the laws hold of all bodies requires an inductive, generalizing step in the argument, because, as Newton concedes, the general truth of the principle can’t be demonstrated; it cannot be proven a priori that the laws hold for all natural bodies. Nevertheless, though their truth may be neither timeless nor universal, the laws stand as timelessly and universally certain propositions that, we can say, are true of, and even true because of, the evidence we have at hand.

4. Conclusion This two-stage reading of Newton’s method gives us a richer sense of how he deploys empirical evidence in the argument for the laws of motion. It is not, as Shapiro suggests, that Newton observed, experimented, and then generalized. And it is not, as ⁶⁷ Newton (2004, 121).

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Cotes claims, that Newton set the ‘phenomena’ as the ‘solid foundation for his brilliant theories.’ Rather, on my account, the foundation is provided by a mathematical account of bodies and motion. Newton begins by reasoning mathematically about mathematically ideal ‘bodies’ in order to identify principles of motions and forces that are beyond rational doubt. Empirical evidence then enters Newton’s argument to bridge what is mathematically certain with what we witness in nature. In other words, as we have seen above, the phenomena are used to ‘prove the truth’ of these principles, i.e., to confirm that the laws can be applied to real physical bodies. Notice that my reading of the laws of motion does not at all diminish Cotes’s point that Newton’s method is informed by a keen awareness that, relative to God, we are ‘small and insignificant’ creatures. For what Newton is doing with his ‘deduction from the phenomena’ is testing the claims of human reason against what God has freely and willfully created. Through observation and experiment, he is establishing that what is certain to us captures what God has made to be the case. As such, the fit between mathematical reasoning and the behavior of real bodies enables Newton to claim that the laws he sets forth are, in light of the best evidence available, the ‘laws by which the supreme artificer willed to establish this most beautiful order of the world.’⁶⁸ In this respect, as Cotes urges, God’s free creation of nature serves, for Newton, as the ultimate standard for truth in natural philosophy. From this richer account of Newton’s method, we also gain a richer account of how his approach to laws of nature compares to Descartes’s. For both, as we have seen, and as fitting of the seventeenth century, the method for identifying laws of nature must be one aimed at achieving an understanding of the world that God freely created. And, as we have also seen, each of them qualifies their attempts to decipher the divine order of things in important ways. With God’s immutable and perfect, yet ultimately incomprehensible, nature as the basis for his investigation of material things, Descartes’s laws of nature are certain and necessary by the standards of human reason, yet true of the world God created only so far as human reason is able to understand. For Newton, in contrast, we begin with a mathematical approach to body and motion, and reason to how bodies that have select quantifiable features necessarily behave. Empirical evidence then serves to connect the mathematical account with the real, physical bodies that populate the natural order such that the laws are true of the world God created only so far as the phenomena reveal. Seeing their notions of truth in this light, a crucial methodological similarity comes into clearer view: To convince us that their laws are true, both Descartes and Newton begin with claims that, by the standards of reason, are to be accepted as indubitable. To be sure, they define ‘the standards of reason’ in different ways: Descartes relies on clear and distinct perceptions, and Newton on mathematical reasoning. Nonetheless, they both ground their attempts to decipher the divine order of things on what is

⁶⁸ Newton (1687/1999, 393).

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humanly and rationally certain, and provide arguments that succeed because they begin with claims that, by their adopted rational standards, are free from doubt. In this respect, both are able to meet the challenge posed by Cotes: They forward programs of natural philosophy that preserve God’s freedom and God’s priority in the order of nature, insofar as their methods of investigation are grounded on the very limitations that define our epistemic position relative to a free and ultimately incomprehensible Creator.⁶⁹

⁶⁹ I had the pleasure of presenting material from this chapter at the University of Minnesota Center for Philosophy of Science, the UC-Irvine Department of Logic and Philosophy of Science, and the Seventh Margaret Wilson Conference, which was held in Flagstaff in June 2016. Several members of these audiences helped me see how I could more effectively craft my discussion of Descartes and Newton, and I am grateful for their willingness to critically engage with my work. I hope this final version addresses at least the more significant concerns that were raised. I am especially thankful for the feedback I received from Zvi Biener, Walter Ott, and Lydia Patton. The comments they provided on the penultimate version of the chapter have made my discussion clearer and more focused than it would have otherwise been.

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4 Leges sive natura Bacon, Spinoza, and a Forgotten Concept of Law Walter Ott

[I]n this Organon of ours we are dealing with logic, not philosophy. But our logic instructs the understanding and trains it, not (as the common logic does) to grope and clutch at abstracts with feeble mental tendrils, but to dissect the powers and actions of bodies and their laws limned in matter [in materia determinatas].¹

The way of laws is as much a defining feature of the modern period as the way of ideas. In one of its forms, it stands as an alternative to the moribund Aristotelian ontology. A world devoid of powers is, in Ralph Cudworth’s memorable phrase, ‘a dead cadaverous thing,’² and explaining why objects in it behave as they do requires appeal to something other than their intrinsic properties. For Descartes and his followers, the way of laws leads to occasionalism. Since bodies cannot obey laws in any but a metaphorical sense, God must be moving them around himself.³ But the way of laws is hardly without its forks. Both before and after Descartes, there are philosophers using the concept of laws to carve out a very different position from his. Descartes’s application of the concept of law in the context of physics has been so influential as to make this competing position all but invisible. Here, I uncover what is, ¹ Francis Bacon, New Organon II.lii: 219–20. (References to the NO are to the Michael Silverthorne translation and are in the following form: Book.aphorism: page number in Bacon 2000a.) Compare Spinoza’s appendix to Part IV of the Ethics, section xxvii: ‘The principal advantage we derive from things outside us—apart from the experience and knowledge we acquire from observing them and changing them from one form into another—lies in the preservation of our body.’ (References to Spinoza’s Ethics and Letter 32 are to the Curley translation in Spinoza (1985). References to other works are to Shirley’s translation (2002) unless otherwise noted. When referring to the Ethics, I first give the part, proposition, and then (S) scholium, (L) lemma, (D) demonstration, (A) axiom, or (C) corollary, if applicable. Thus ‘2p13L1’ refers to Part II, proposition 13, Lemma 1. The Latin text is from the Gebhardt edition of 1925.) ² Cudworth (1837, vol. 1, 209). ³ I make this argument at length in my (2009).

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as far as I can tell, an unappreciated sense of ‘law,’ one that is entirely disconnected from God or God’s will. This branch of the way of laws antedates Descartes’s work in the person of Francis Bacon, and post-dates it in that of Baruch Spinoza. Both philosophers reject the portrait of the world as an inanimate lump, dependent on God to shove it about. If I am right, Bacon and Spinoza tread a different path, one that is actively opposed to the inert mechanical conception of nature.⁴ Roughly, a law of a thing’s nature is the set of powers that defines that nature. Different natures have (or just are) different laws, and they can restrain and interfere with each other. Descartes’s introduction of leges exploits one facet of the legal analogy: the sense in which a prince or king can lay down a set of rules his subjects must obey. In just the same way, Descartes’s God gives extended substance its marching orders. Bacon and Spinoza point to a different facet of the legal analogy: the sense in which laws describe what must happen in a variety of different circumstances. To fully grasp a nature like heat is to learn all of the conditionals that are true of it in virtue of its powers, and those of the natures it can encounter. By pointing to this feature of the civil law, Bacon and Spinoza decline the seventeenth century’s invitation to move beyond powers and into the cadaverous world of mechanism. Fully grasping this alternative, I argue, requires us to rethink Spinoza’s metaphysics of causation. I begin by unearthing the concept in the work of Francis Bacon, who, no less than Spinoza, thinks of laws as being ‘inscribed’ in ‘things [a legibus in iis rebus inscriptis], as in their true codes, according to which all singular things come to be, and are ordered.’⁵ I then turn to Spinoza’s Letter 32, which, I argue, moves from what we might call the ‘human image’ of the physical world to its true metaphysical image. The following section explores the Ethics, which moves in the opposite direction: we begin with the skeletal metaphysical truth and then reconstruct the world as it appears to us in experience. Finally, I draw out the consequences of this reading for the rest of Spinoza’s view.

1. At first sight, Bacon’s thoughts on laws, even with a single text like The Novum Organon (NO), are a disappointing mishmash of unanalyzed jargon. He tells us that substances have laws,⁶ that laws govern acts,⁷ that laws are forms, which in turn are forms of natures,⁸ and, to make matters worse, that forms derive natures ⁴ My focus is on the non-human world. I am concerned with laws as they play a role in physics, as opposed to morality or the law. Donald Rutherford (2010) helpfully distinguishes between what he calls Type-I laws, which ‘express natural necessity,’ and Type-II laws, which depend on human decisions and have prescriptive force. Rutherford mounts an ingenious argument designed to show that Type-II laws are not an irreducibly novel category of laws but instead are ‘principles of natural necessity grounded in the inherent power of the human mind’ (2010, 167). Thus Type-I laws are fundamental; Type-II can be seen as a special case of natural necessity as it is worked out in the sphere of human agency. Though nothing in my analysis depends upon it, Rutherford’s reading strikes me as plausible and consistent with what I say here. ⁵ Treatise on the Emendation of the Intellect, }101 (1985, 41). ⁶ NO II.iv: 104. ⁷ NO I.li: 45, II.ii: 103, II.v: 106. ⁸ NO II.ii: 102–3.

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from essences.⁹ Sometimes he speaks as if laws govern simple natures like heat; at other times, as if the laws just were the simple natures; and in one startling sentence, he does both at once: ‘when I speak of forms, I mean nothing more than those laws and determinations of absolute actuality which govern [ordinant] and constitute any simple nature, as heat, light, weight, in every kind of matter and subject that is susceptible of them.’¹⁰ Before sorting through all of this, we have to remove a barrier to understanding Bacon. There is a strong temptation to fold Bacon into the wave of mechanical philosophers that was sweeping Europe.¹¹ For such philosophers, Aristotle’s four causes are gradually whittled down to one: the efficient cause. All other kinds of explanation are either epistemically out of reach (the final cause, since it is impossible to know the mind of God),¹² or part of the detested hylomorphism of the scholastics (the formal and material causes).¹³ On one isotope of this sort of view, found in Robert Boyle and John Locke, the ultimate hope of natural philosophy is uncovering the micro-level structures that explain the efficient causal interactions among bodies.¹⁴ Reading such views back into Bacon is a mistake. Bacon opens Book II of the NO by endorsing the four causes of Aristotle. His reservations are actually focused on the efficient and material causes: they are ‘perfunctory, superficial things, of almost no value for true, active knowledge.’¹⁵ Knowing what something is made of, or what produces such-and-such an effect, falls short of ‘true’ knowledge partly because it does not help one reproduce the effect, which is the aim of ‘active’ knowledge. Bacon is well aware this emphasis on forms will sound odd, since he has made fun of the scholastics’ empty jargon in Part I. Here is his replacement doctrine of forms: For though nothing exists in nature except individual bodies which exhibit pure individual acts (‘actus puros’) in accordance with law, in philosophical doctrine, that law itself, and the investigation, discovery and explanation of it, are taken as the foundation both of knowing and doing. It is this law and its clauses which we understand by the term Forms, especially as this word has become established and is in common use.¹⁶ ⁹ NO II.iv: 104. ¹⁰ NO II.xvii: 128. ¹¹ As Gaukroger (2001) tends to do. ¹² As Descartes tells Gassendi, ‘[w]e cannot pretend that some of God’s purposes are more out in the open than others; all are equally hidden in the inscrutable abyss of his wisdom’ (AT VII 375/CSM II 258). See also Meditation 4, AT VII 55/CSM II 39. (References to Descartes are to the Cottingham, Stoothoff, and Murdoch translation (CSM) and to Adam and Tannery’s edition of Descartes’s work (AT).) ¹³ For the contraction of the four causes into efficient cause, see esp. Vincent Carraud (2002). ¹⁴ Locke of course is skeptical that this aim could be achieved; nevertheless, when he allows himself to speculate about an idealized science, what he envisions fits my reading. For example, Locke writes, ‘If we could discover the Figure, Size, Texture, and Motion of the minute Constituent parts of any two Bodies, we should know without Trial several of their Operations one upon another, as we do now the Properties of a Square, or Triangle . . . The dissolving of Silver in aqua fortis, and Gold in aqua Regia, would be, then, perhaps, no more difficult to know, than it is to a Smith to understand, why the turning of one Key will open a Lock, and not the turning of another’ (IV.iii.25, in Locke 1689/1975, 556). ¹⁵ NO II.ii: 102. The final cause is ‘a long way from being useful’ except in cases of human action. ¹⁶ NO II.ii: 103. Similarly, Newton says that he takes principles like cohesion and gravity not as ‘occult qualities, supposed to result from the specific forms of things, but as general laws of nature, by which the things themselves are formed’ (2004, 137).

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Some commentators, such as Stephen Gaukroger, try to read such passages away, as if Bacon really meant to be talking about matter and its arrangement rather than law.¹⁷ Such a move is unmotivated if, as I shall argue, we can in fact make good sense of these passages.¹⁸ When a body engages in an ‘actus puros,’ it acts in a way that is not interfered with or distorted by another body.¹⁹ Such activities will of course be the most informative about a given nature. Now, most bodies will be composites of the simple natures, such as heat. So the task of natural philosophy is to separate out these simple natures and learn their laws. Bacon conceives of laws as including clauses that detail how something possessed of the form in question will behave under suchand-such circumstances, just as a clause in statute law details how the law is meant to apply to particular cases.²⁰ Note just how foreign this way of thinking is to its main competitors, whether Aristotelian or Cartesian. The scholastic notion of a form as a principle of explanation in its own right is rejected; such forms are ‘figments of the human mind,’ unless by ‘form’ one just means ‘the law of act or motion.’²¹ And unlike the Cartesian conception, there is no suggestion that there might be just one set of laws governing the whole universe: there are irreducibly different natures out there, each with its own set of laws.²² What is more important, Bacon’s conception of laws is thoroughly bottom-up. So far, Bacon’s system seems unnecessarily crowded: we have natures, forms as laws, and even essences, all vying for the role of explanans. Throughout Book II, Bacon cleans up his system by collapsing some of these into others.²³ Aphorism iv tells us that form and nature always go together: when the form of heat is present,

¹⁷ See Gaukroger (2001, 140 f.) for an ingenious attempt to read away the textual evidence. ¹⁸ For other places in NO where Bacon treats forms as laws, see, e.g., I.li: 45, II.v: 106, II.xvii: 128. ¹⁹ As Jonathan Bennett suggests in his commentary on the passage; see his . For another use of the phrase, see NO I.li: 45. ²⁰ Jardine and Silverthorne note the analogy with statute law in Bacon (2000a, 103, n. 2). ²¹ I.li in Bacon 2000a, 45. ²² Gaukroger notes that Bacon sometimes speaks of fundamental and universal laws. For example, at NO II.v, Bacon writes that the inquiries into such things as the voluntary motion of animals ‘are concerned with compound natures, or natures which are joint members of a structure; and they have regard to special and particular habits of nature, not the fundamental and common laws which constitute Forms’ (NO II.v: 106). Gaukroger reads this as an endorsement of ‘general laws of nature’ (2001, 141); Bacon’s point, in part, is that it is not enough to know the general laws of nature in order to explain the motions of animals. But there is no suggestion in the text that there is a single law, or set of laws, that governs all of nature, as Gaukroger thinks. When Bacon talks about the ‘fundamental and common laws which constitute Forms,’ he is referring to the dispositions that constitute simple, as opposed to composite, natures. The contrast Bacon draws is between simple and composite natures, not between specific laws (say, of mechanics) and more general ones. ²³ In fact, Bacon goes further and identifies a thing with its form. See, e.g., NO II.xiii: 119, where Bacon writes, ‘The Form of a thing is the very thing itself. And a thing does not differ from its form other than as apparent and actual differ, or exterior and interior . . . and hence it follows that a nature is accepted as a true form unless it always decreases when the nature itself decreases and increases when the nature increases.’

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so is the nature, and when the form is absent, so is the nature. By the time he reaches xvii, Bacon has decided to identify these. When we speak of forms, we mean simply those laws and limitations of pure act which govern [‘ordinant’] and constitute a simple nature, like heat, light, or weight, in every kind of susceptible material and subject. The form of heat therefore or the form of light is the same thing as the law of heat or the law of light, and we never abstract or withdraw from things themselves and the operative side. And so when we say (for example) in the inquiry into the form of heat, Reject rarity, or, rarity is not of the form of heat, it is the same as if we said, Man can superinduce heat on a dense body, or on the other hand, Man can take away heat or bar it from a rare body.²⁴

Forms, then, are nothing more than laws. But how can Bacon at once identify the laws with the simple nature and claim that the laws ordinant that nature? Clearly he can’t have it both ways. The problem here is an ambiguity in Bacon’s notion of a nature. In its usual use, ‘a nature’ is a non-dispositional feature or property that we can experience, such as light or heat.²⁵ Natures stand in need of explanation. This explanation might take two forms: one might give the material cause, which is the underlying stuff and its organization. In this sense, the nature heat is expansive motion. But as we have seen, Bacon regards this as being of relatively little interest to the natural philosopher. What counts is the formal cause, that is, the dispositions or powers. From a contemporary perspective, it is tempting to go further and claim that the dispositions described by the forms/laws are grounded in the structure of the matter in question. And that may very well be what Bacon has in mind, though it is not strictly speaking there in the text. A word for a nature like ‘heat’ might equally well refer to (i) the categorical property we experience, (ii) the forms/laws or dispositions that go along with that property, or (iii) the ultimate micro-structure that grounds those laws.²⁶ So when Bacon says that forms/laws govern or organize a nature, he means nature in sense (iii), the micro-structure. And when he says that forms/laws constitute a nature, he means nature in sense (i), the property we experience. And, trivially, when he says that forms/laws are natures, he is using the term in sense (ii). It is annoying but entirely natural for Bacon to slide from one of these senses to another. From here on, I’ll use nature in sense (i) only, as the explanandum rather than explanans.

²⁴ NO II.xvii: 128. For consistency, I am rendering ‘ordinant’ as ‘govern’ rather than ‘organize,’ as this translation of NO has it. ²⁵ Here I ignore the difference, which Bacon registers, between heat as felt and heat as it is in the world. See NO II.xx: 131, where Bacon claims that ‘[h]eat as felt is a relative thing . . . and it is rightly regarded as merely the effect of heat on the animal spirit.’ Throughout, I use ‘heat’ in Bacon’s second sense, that is, as the non-relative nature that exists independently of its being felt. ²⁶ For (iii), see esp. NO I.li: 45, where Bacon recommends the study of ‘matter, and its structure (schematismus).’

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There is a final layer of complexity. It is important, I think, to take Bacon seriously when he elects to open Book II of the NO by saying that ‘[t]he task and purpose of human Science is to find for a given nature its Form, or true difference.’²⁷ Nor is this an isolated claim; Bacon goes on to say that laws/forms act as differentiae which contract an ‘essence’ into a determinate nature.²⁸ A true form ‘derives a given nature from the source of an essence which exists in several subjects.’²⁹ But what can this mean? Since forms/laws are differentiae, Bacon must be using ‘essence’ to mean ‘genus,’ a not unnatural usage. So multiple natures can share an essence or genus; we need to find what sets one off from the other. And that something is the form/law that characterizes it. Heat, for example, is not just motion, an essence that can be shared across many natures. Heat is motion that behaves in a particular way, namely, it tends to expand. This is why forms are the verae rerum differentiae. So we now have a nature (e.g., heat) whose form just is its laws, that is, the set of dispositions that characterize its behavior. Finding the form amounts to true and active knowledge because knowing those dispositions tells you how to bring it about and control it. Although there is nothing in the physical world beyond bodies and their acts, the goal of science is the discovery of forms/laws, not of the microstructures that underwrite them. These forms are captured in propositions that describe possible states of affairs and could equally be cast as conditionals. What makes these conditional or modal statements true will be the powers of the objects that figure in them. What grounds these dispositions, in turn, will presumably be matter and its structure. While Bacon is often cast as a harbinger of mechanism, he retains something vitally important of Aristotelianism: the emphasis on the explanatory (and predictive) power of power.³⁰ Even if, as Gaukroger maintains, for Bacon ‘the ultimate ingredients’ of things are matter structured in such-and-such a way, Bacon’s notion of the law of a given nature remains the ultimate ingredient of explanation and the target of scientific inquiry.³¹ Bacon’s view can be illuminated by contrasting it with Descartes’s. For Descartes, the laws of nature are captured by propositions such as ‘each thing . . . always remains in the same state, as far as it can, and never changes except as a result of external causes.’³² Such propositions are necessarily true, at least so long as the nature and will

²⁷ NO II.i: 102. ²⁸ ‘[A] true form is such that it derives a given nature from the source of an essence which exists in several subjects’ (II.iv in Bacon 2000a, 104). Bacon identifies the forms with these differentiae; he speaks of the ‘forms or true differences of things (which are in fact laws of pure act)’ (I.lxxv in Bacon 2000a, 62). ²⁹ NO II.iv: 104. ³⁰ ‘The task of science is to find for a given nature its Form, or true difference, or causative nature, or the source of its coming-to-be’ (NO II.i: 102). ³¹ See Gaukroger (2001, 140). ³² Principles II.37, AT VIIIA 62/CSM I 240–1.

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of God remain constant.³³ Descartes’s notion of laws is ‘top-down’ in the sense that the laws flow from the nature of the divine being, not the natures of the things that ‘obey’ them.³⁴ Nothing could be further from Bacon’s unrepentantly bottom-up approach. Someone might well argue that the difference between Descartes and Bacon is merely verbal: if Bacon wants to use ‘law’ to mean power or disposition, he is welcome to it. The orthographic similarity of the terms about which they disagree does not necessarily mark disagreement. I think this objection misses the point. My claim is that Descartes and Bacon are both working with a recognizably legal metaphor. For Descartes, what counts is the arbitrariness of human laws, which are set down at will by the lawgiver. For Bacon, what counts is the structure of a statute law, with its extensive clauses designed to cover every possible case that might arise. Although either path along the way of laws can be consistent with the ontology of mechanism, they nevertheless point in different directions: top-down (Descartes) and bottom-up (Bacon). It remains now to see which path Spinoza will tread.

2. We know from his correspondence that Spinoza read Bacon carefully. Even if he abuses Bacon when it comes to the nature of the intellect and the human mind, he also refers one of his correspondents to Bacon on just those issues.³⁵ And he credits Bacon, alongside Descartes, with having proved that all qualities depend on matter and motion.³⁶ Bacon takes the first step away from the Aristotelian view when he trades in the scholastics’ vast panoply of natural kinds for a comparatively small number of simple natures.³⁷ There is no way to reduce one of these simple natures to another; but learning this manageably small set of laws would amount to learning nature’s alphabet, the first step to deciphering its words. Spinoza, of course, wants to go ³³ Descartes’s thinking on modality, and the sense he gives to terms like ‘necessary,’ is obscure and controversial. My point here is only that, since the laws of nature follow from God’s nature and will, they hold in any world where those features of God obtain. ³⁴ See chapter 1 of my (2009) for more on the top-down/bottom-up distinction. There, I argue at length, pace Hattab (this volume) and others, that Descartes’s view of body-body causation has to be occasionalist. I won’t repeat those arguments here. ³⁵ See Letter 37 (1666) in Spinoza (2002, 861). ³⁶ In Letter 6 (1661) (2002, 771 f.) Spinoza says that Lord Verulam, along with Descartes, has proved that all tangible qualities depend on motion, shape, and other mechanical states. ³⁷ In The Advancement of Learning, Bacon writes, ‘In the same maner to enquire the forme of a Lyon, of an Oake, of Gold; Nay of Water, of Aire, is a vaine pursuite: But to enquire the formes of Sence, of voluntary Motion, of Vegetation, of Colours, of Gravitie and Levitie, of Densitie, of Tenuitie, of Heate, of Cold, & al other Natures and qualities, which like an Alphabet are not many, & of which the essences (upheld by Matter) of all creatures doe consist; to enquire I say the true formes of these is that part of METAPHISICKE, which we now define of ’ (1605/2000b, 84).

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much further. The important point is that he is going much further in the same direction as Bacon, not Descartes.³⁸ Part of what makes interpreting Spinoza so difficult is the transition he asks us to make, often without an explicit invitation, between the human image—the world seen from the limited, human point of view—and the metaphysical image. The world that we live in, and which natural philosophy is suited to investigate, is replete with bodies that have distinct natures or essences (such as man, horse, and insect).³⁹ Seen from the point of view of eternity, however, the physical world is a single substance that has only one nature, namely, extension.⁴⁰ In my view, Spinoza’s Letter 32 (‘The Worm in the Blood’) enacts for us the transition from the manifest image to the metaphysical image, where bodies constitute a single super-individual, what Spinoza elsewhere calls ‘the face of the entire universe.’ Spinoza’s Ethics takes us in precisely the opposite direction. There, Spinoza begins with the metaphysical image and goes on to show how the human image can be reconstructed. There is no question but that the metaphysical image is fundamental. The challenge for Spinoza is to explain why the world as we experience it seems so very different. In this section, I focus on Letter 32. Here we travel from a human image, one that is recognizably Aristotelian, through a Baconian view, and end up finally with the world as it is in itself. Letter 32 illustrates how one might start by thinking of the world in terms of powers or natures and end with an ontology that has room for only one nature and the patterns it produces. In this letter, Spinoza responds to a question from Oldenburg: ‘how [do] we know how each part of Nature agrees with the whole to which it belongs and how it coheres with the others’?⁴¹ Whatever Oldenburg means by ‘coherence,’ Spinoza stipulates that ‘[b]y the coherence of parts, then, I understand nothing but that the laws or {sive} nature of the one part so adapt themselves to the laws or nature of the other part that they are opposed to each other as little as possible.’⁴²

³⁸ In less technical works, Spinoza sometimes uses ‘lex’ in a very Cartesian way. For example, in the Tractatus Theologico-Politicus, he writes that ‘the universal laws of Nature according to which all things happen and are determined are nothing but God’s eternal decrees, which always involve eternal truth and necessity’ (2002, 417). A bit later, however, he makes it clear that he is speaking analogically: ‘Still, it seems to be by analogy that the word law is applied to natural phenomena, and ordinarily “law” is used to mean simply a command which men can either obey or disobey, inasmuch as it restricts the total range of human power within set limits and demands nothing that is beyond the capacity of that power’ (2002, 427). But Spinoza’s real view is even more distant from the Cartesian picture, since of course he rejects the whole notion of a God who stands outside of nature and sets out his ‘decrees.’ ³⁹ See, e.g., the preface to Part IV of the Ethics. ⁴⁰ The one substance of course has an infinite number of attributes, of which we know only two, thought and extension. ⁴¹ Spinoza (1994, 82). ⁴² Spinoza (1994, 82). Note that Curley’s italicized ‘or’ translates sive, which indicates an equivalence rather than an alternative.

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Spinoza begins by insisting that we do not know how this happens, though we do know that it does. Even if there is only one substance, the natural world still seems, from the human point of view, to be divided into parts: Concerning wholes and parts, I consider things as parts of some whole insofar as the nature of the one so adapts itself to the nature of the other that so far as possible they are all in harmony with one another. But insofar as they are out of harmony with one another, to that extent each forms an idea distinct from the others in our mind, and therefore it is considered as a whole and not a part.⁴³

Whether a mereologically neutral ‘thing’ is a whole or a part is a function of harmony and coherence. When we consider two things as two, that is, as two wholes, that is because their disharmony produces two distinct ideas in the mind. Now we know that disharmony is ultimately an illusion, as is the two-ness of the two, or the n-ness of the n. Nevertheless, at this initial stage, we attribute distinct and competing natures and laws, that is, distinct powers, to distinct parts of the natural world. We can now confront the central puzzle of Letter 32: what could Spinoza mean by speaking of the laws of a thing’s nature? Once we see the Baconian context, and construe laws as powers, Spinoza’s text comes into focus. The laws of a thing’s nature just are the powers it has. This is why Spinoza can speak indifferently of laws or natures and laws of natures. Either way, what makes heat heat is what it does in suchand-such circumstances. Spinoza tells us that the laws of different objects ‘restrain’ each other.⁴⁴ Such a pronouncement is incoherent on the Cartesian understanding of laws: the laws are universal and unitary. But if laws are powers, it makes perfect sense: the powers of distinct objects can compete against each other. The blood might be disposed to move in a certain direction, but this disposition can be interrupted, if you like, by the presence of competing causes. In just the same way, a match can be disposed to light when struck, and yet not light in an oxygen-free environment. Any given event, then, will be the result of the interplay among the laws or dispositions of the objects concerned. Such a view has some striking affinities with Aristotelian thought. The nature of fire, on such a view, is a disposition to burn things. But whether anything actually is burned is a function partly of the passive powers possessed by the objects with which fire comes into contact. An Aristotelian science is occupied with discovering these natures or powers and mapping out their interactions. To illustrate this stage, Spinoza introduces an analogy. He asks us to imagine things from the point of view of a little worm living inside the veins of another creature. To us, the particles that make up blood (lymph and chyle, for example) ⁴³ 1994, 82–3. ⁴⁴ ‘There are a great many other causes which restrain the laws of the nature of the blood in a certain way, and which in turn are restrained by the blood, it happens that other motions and other variations arise in the particles of the blood’ (1994, 83).

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are parts of the blood. But to a worm swimming in the blood, endowed with the capacity to distinguish these particles, lymph and chyle are not parts but wholes. Spinoza writes, [The worm] would live in this blood as we do in this part of the universe, and would consider each particle of the blood as a whole, not as a part. Nor could it know how all the parts of the blood are restrained by the universal nature of the blood, and compelled to adapt themselves to one another, as the universal nature of the blood requires, so that they harmonize with one another in a certain way.⁴⁵

The worm, it seems, is missing out on two things. First, it cannot see how the particles, which it regards as wholes, are themselves parts of the blood and so are ‘restrained by’ the nature of the blood. The explanation for how the lymph particle behaves involves the nature of the blood as a whole. Just as we do (at least before we come to realize the truth), the worm sees every variation in powers as a mark of a new whole, a new and complete object, endowed with a novel set of powers. The Aristotelian watching fire burn paper has done the same thing. Second, the worm has no way of knowing that there is anything beyond the blood. And just as the particles of the blood ‘adapt themselves’ to one another, so the blood itself is adapted to or harmonizes with the body of which it is a part. And of course beyond the human body itself there are many other things that affect the blood, whether directly or indirectly. Spinoza sums up the point of his example in this way: Now all bodies in Nature can and must be conceived as we have here conceived the blood, for all bodies are surrounded by others, and are determined by one another to existing and producing an effect in a certain and determinate way, the same ratio of motion to rest always being preserved in all of them at once, that is, in the whole universe. From this it follows that every body, insofar as it exists modified in a certain way, must be considered as a part of the whole universe, must agree with the whole to which it belongs, and must cohere with the remaining bodies. And since the nature of the universe is not limited, as the nature of the blood is, but is absolutely infinite, its parts are restrained in infinite ways by this nature of the infinite power, and compelled to undergo infinitely many variations.⁴⁶

What we call ‘bodies’ are not wholes but parts of the entire universe. An infinite mind would stand to all bodies as we stand to the blood. Our grouping of the modes we experience into bodies is a function of our benighted epistemic state; what is real is only the single substance.⁴⁷ And our belief that they exhibit genuinely different

⁴⁵ 1994, 83. ⁴⁶ 1994, 84. ⁴⁷ Spinoza’s thoughts here are very much in line with 2p13L7, where he argues that ‘the whole of nature is one individual, whose parts, that is, all bodies, vary in infinite ways, without any change of the whole individual.’ For more on how the parts can vary while the whole stays the same, see esp. Tad Schmaltz (1997). For our purposes now, the important part is the notion that what we call bodies are parts of a single super-individual.

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powers is also an illusion: since all (physical) modes are modes of extension, they share a nature, and their behavior is fully explicable by that single nature. Thus Letter 32 presents a pathway by which one might navigate from a naïve initial position—roughly the Aristotelian view—to a metaphysically sound, Spinozistic view. We begin with the claim that the physical world is exhaustively characterized by a single attribute, viz., extension. For the various Aristotelian natures, with all their attendant and competing powers, we substitute a single nature and a single set of dispositions and powers.⁴⁸ One can then identify the laws of nature with that nature. And if one takes the further step and treats bodies not as substances but as modes, then physical events can be seen as the working-out of a single nature of a single substance, across merely epistemically distinct bodies.

3. If Letter 32 moves from the human to the metaphysical image, the Ethics moves in just the opposite direction: Spinoza begins in Part I with the core notions of his metaphysics. It is left to the other parts to restore, bit by bit, the human image. For our purposes, it will thus be better to read the Ethics backwards, working from the roughly Aristotelian picture back to the true metaphysical one. In Parts III and IV, we find Spinoza happily attributing different essences to man, insects, and horses, even though we know he cannot really be serious. The next level is exhibited by the so-called Physical Digression of Part II. There, Spinoza is closest to Bacon, winnowing the lineup of distinct essences down to a limited number of forms such as hard, soft, and fluid. Finally, when we turn back to Part I, we find all of this talk falls away, and the one substance and the laws of its nature, that is, its power, emerge as the fundamental notions of Spinoza’s metaphysics. Salted throughout the final parts of the Ethics, one can find unapologetically Aristotelian language. Perhaps the most important comes in the preface to Part IV, where Spinoza writes, [W]hen I say that someone passes from a lesser to a greater perfection, I do not understand that he is changed from one essence, or form, to another. For example, a horse is destroyed as much if it is changed into a man as if it is changed into an insect.

Aquinas would not have blushed to write this. Forms are migration barriers; nothing can pass from one to another without being destroyed. At this human level, forms and essences are the same thing. What that thing is remains obscure. In a way, that is no surprise, for this everyday notion of form/essence must be refigured as we ascend closer to the truth of things. ⁴⁸ I am not, of course, suggesting that Letter 32 invokes Aristotelian forms. Instead, it is clearly their mechanical substitutes Spinoza has in mind: the exercises of the dispositions of the worm and the blood are to be understood exclusively in terms of motion-and-rest.

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Aristotelian forms have to be analyzed away. In Part III, Spinoza gives us the tools to do so: 3p6: ‘Each thing, as far as it is in itself, endeavors to persist in its own being.’⁴⁹ 3p7: ‘The striving by which each thing strives to persevere in its being is nothing but the actual essence of the thing.’ The conatus or striving is not something added on to a thing, or something it has only because God moves it about in a certain way. It is the essence of the thing. Since physical things are not substances but modes, their essences are in the end modifications of the attribute of extension. The terminology is tangled but the point should be clear: that bodies strive to persist in their own being is part of what it is to be a body. To make sense of this, we have to invoke Part II’s Physical Digression (2p13). We know from the Definitions at the beginning of that part that an essence is something the positing of which is both necessary and sufficient for the thing’s being posited. The Physical Digression reveals that what individuates composite bodies is their differing ratios of motion and rest.⁵⁰ If we identify what makes an individual that individual with its essence, we can say that the essence of, say, a horse or a tree is its striving to maintain the same ratio of motion and rest.⁵¹ Presumably, if it fails in this task, it then disintegrates into its constituent simple bodies. What makes 2p13 so hard to grasp is the fluidity of Spinoza’s jargon. At times, he appears to use ‘form’ or ‘nature’ interchangeably, where both mean Aristotelian forms.⁵² The Physical Digression, then, is meant in part as a respectably mechanist reduction of Aristotelian forms to ratios of motion and rest. So far, so good. But there is also reason to think Spinoza is using these terms to mean something like macrolevel property or state. Hard, soft, and fluid are forms, according to Spinoza, and ‘natures’ according to Bacon.⁵³ Unlike Aristotelian essences, things can gain and lose these forms without being destroyed.⁵⁴ We have not yet reached the metaphysical image. That is the work of Part I. Not surprisingly, or so I shall argue, Spinoza uses the notion of law to make the transition.

⁴⁹ Here again I depart from Curley, who renders 3p6 thus: ‘Each thing, as far as it can, strives to persevere in its being by its own power.’ As Deborah Brown pointed out to me, nothing in the text justifies Curley’s use of ‘power’: ‘Unaquaeque res, quantum in se est, in suo esse perseverare conatur.’ There is also a philosophical problem in the interpretation of this proposition. As it stands, it allows that things sometimes don’t strive to persevere in being; but 3p7 makes such striving the essence of a thing. Michael Della Rocca (1995, 200) suggests we read 3p6 as saying that each thing will persevere in being, so far as it can. It’s the persevering, not the striving, that external forces can prevent. ⁵⁰ ‘Bodies are distinguished from one another by reason of motion and rest, speed and slowness, and not by reason of substance’ (2p13L1). ⁵¹ As Nadler (2006, 195) helpfully points out. ⁵² See esp. 2p13L6 and L7. ⁵³ See Ethics 2p13 and NO II.xxv: 140. ⁵⁴ Spinoza speaks of the experience and knowledge we derive ‘from observing [things outside us] and changing them from one form into another’ (appendix to Part IV, section xxvii).

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After establishing that God exists (1p11) and is the only substance (1p14), Spinoza has to explain how a single substance can be responsible for the diversity we see around us. His answer is precisely what you get when you combine Bacon’s understanding of laws with monism: 1p16: ‘From the necessity of the divine nature there must follow infinitely many things in infinitely many modes.’ 1p17: ‘God acts from the laws of his nature alone, and is compelled by no one.’ Just as the definition of a triangle entails properties like having angles that sum to 180 degrees, so a nature like extension will logically imply an infinite number of properties. If among these properties are finite modes, we have some notion of how the attribute of extension might generate bodies, i.e., finite modes. In these propositions, we begin to see just how far Spinoza is from the Cartesian concept of extension. Spinoza’s extension is not a cadaverous thing awaiting a divine shove; it is active of itself.⁵⁵ Note how Spinoza puts it when it comes time to say how God acts: he acts ‘solus suae naturae legibus,’ from the laws of his nature alone. As in Letter 32, Spinoza’s point is that there is no competing substance with a different nature, whose laws (that is, powers or dispositions) could interfere with God’s. Without the Baconian background, it is impossible to read 1p17 as anything other than a long printer’s error. Later on in the Ethics, Spinoza again uses the notion of law to knit together the human and metaphysical images: Nature is always the same, and its virtue and power of acting [‘virtus et agendi’] are everywhere one and the same, that is, the laws of nature and the rules according to which all things happen,⁵⁶ and change from one form to another, are always and everywhere the same. So the way of understanding the nature of anything, of whatever kind, must also be the same, namely, through the universal laws and rules of Nature.⁵⁷

⁵⁵ Tschirnhaus (Letter 80) pressed Spinoza on just how extension by itself could entail all the infinity of finite modes that are supposed to follow from its nature (E1p16). In answer, Spinoza distances his concept of extension from that of Descartes: ‘[F]rom Extension as conceived by Descartes, to wit, an inert mass, it is not only difficult, as you say, but quite impossible to demonstrate the existence of bodies. For matter at rest, as far as in it lies, will continue to be at rest, and will not be set in motion except by a more powerful external cause. For this reason I have not hesitated on a previous occasion to say that Descartes’s principles of natural things are of no service, not to say quite wrong’ (Letter 81). In Letter 83, Spinoza adds that ‘matter is badly defined by Descartes by means of Extension’ and that ‘it must necessarily be explicated by means of an attribute that expresses eternal and infinite essence’ (Quoted in Schmaltz 1997, 220.) That doesn’t take us very far toward understanding Spinoza’s notion of extension, but it does mark his view off from Descartes’s. ⁵⁶ Here I depart slightly from Curley’s translation. As he has it, Spinoza says that ‘the laws and rules of nature, according to which all things happen, are everywhere the same.’ I can understand the desire to avoid repetition, but it might be significant that the Latin phrase is ‘naturae leges et regulae secundum quas omnia fiunt,’ that is, ‘the laws of nature and the rules according to which all things happen.’ ⁵⁷ Preface to Part III. Cp. Treatise on the Emendation of the Intellect, section 101.

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Change in form is to be attributed to the laws of the one nature that is instantiated in the physical world. And since there is but the one nature, there is only one set of laws, that is, one set of dispositions or powers. The contemporary ear is tempted to seize on Spinoza’s talk of ‘the universal laws and rules of nature’ and rush to one of the contemporary analyses of laws of nature on offer. That would be a mistake. The question is, in which direction does the reduction run: is it from laws to powers, or the other way around?⁵⁸ What we have already established should make the choice clear: talking of laws or rules is just talking of powers or dispositions.⁵⁹ And just as 1p16 and 1p17 claim, there is only one nature in nature, namely, extension; hence its powers are uniform and can never encounter a competing nature that might disrupt it. But it is not just his overt claims that force us to read Spinoza’s laws as powers or dispositions. His practice, when it comes time to prove propositions Descartes would happily call leges, also supports this reading. In the Physical Digression of 2p13s, Spinoza ‘premise[s] a few things concerning the nature of bodies.’ What follows are a set of axioms and lemmas; Spinoza never calls them leges. These propositions follow from the nature of body. Consider how Spinoza proves Lemma 3, which states, in part, that a body that moves or is at rest must be determined to do so by another body, and so on to infinity. Bodies, Spinoza argues, are individuated by motion or rest (2p13L1).⁶⁰ Since each finite mode has another finite mode as its cause (1p28), and since we have ruled out cross-attribute causation, a particular body’s motion-or-rest must have as its cause another body. Given the way bodies are individuated, we can infer that this body itself must be in a state of motion-or-rest. And we can then run the same argument: there must be yet another body to account for its motion-or-rest, and so on ad infinitum. By contrast, Descartes offers to prove a closely related proposition by appeal to God’s nature and activity. His first law in the Principles states that ‘each thing, in so far as it is simple and undivided, always remains in the same state, as far as it can [‘quantum in se est’], and never changes except as a result of external causes.’⁶¹ Putting it that way suggests that there is something in the body that explains its

⁵⁸ Curley reads the passage as indentifying ‘the laws of nature and nature’s virtue and power of acting’ (correspondence, reported in Melamed (2009, 25); cp. Curley (1969, 49) and (1988, 42–3)). On Curley’s view, nature’s agency amounts to nothing more than its being rule-governed. That is at best backwards and at worst nonsensical (since Spinoza does not have the Cartesian, governing conception of laws). ⁵⁹ See Melamed (2009, 25–6) for a selection of possible competing readings of this obscure passage. Melamed modestly concludes that ‘the textual source appears too equivocal to support the bold suggestion that God is the most general law (or principle) of nature’ (26). ⁶⁰ Note that we are dealing with simple bodies (i.e., those individuated solely by motion-or-rest), not composite bodies—which are only introduced at the end of A2. It is also important to note that the natures of body that Spinoza speaks of in A3 and following are not Aristotelian natures but merely hard, soft, or fluid characteristics, all of which are explained by mechanical means. ⁶¹ Principles II.37 (AT VIIIA 62/CSM I 240–1).

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persistence in the same state. But Descartes explicitly says that the reason this law obtains lies in ‘the immutability and simplicity of the operation by which God preserves matter in motion.’⁶² What is moving is not nearly as important to the law as by whom it is being moved. When Spinoza uses the same phrase (‘quantum in se est’) in 3p7, he might well be self-consciously inverting Descartes’s position.

4. By now, it should be clear that Spinoza has nothing like a Cartesian concept of leges naturae, nor the contemporary concept of laws of nature that descends from it. At times, in writings less technical than the Ethics, Spinoza is happy to speak of laws as if they were divine decrees. But when he is being careful, it is clear that laws of nature are powers or dispositions. What might justly be called the dominant reading of Spinoza on laws, however, imports a contemporary conception of laws as general facts that govern the behavior of objects in the universe. In their different ways, Edwin Curley, Jonathan Bennett, and Jon Miller subscribe to what we might call this ‘general facts’ view.⁶³ It is worth briefly canvassing this view as a foil to bring out what is distinctive about my Baconian interpretation. The best way to see the appeal of the general facts view is by considering the puzzle it is meant to solve. Spinoza distinguishes between finite and infinite modes. Among the latter, we can also distinguish mediate and immediate infinite modes, depending on their relationship to the attribute in question. The immediate infinite mode under attribute of extension is, according to Letter 64, motion and rest; the mediate infinite mode is said, in a notoriously obscure phrase, to be facies totius universi, or the face of the entire universe. The challenge is to see how any of these infinite modes can be related to finite modes. All modes whatsoever must follow from the infinite modes: 1p23: ‘Every mode which exists necessarily and is infinite has necessarily had to follow either from the absolute nature of some attribute of God [an immediate infinite mode] or from some attribute, modified by a modification which exists necessarily and is infinite’ [a mediate infinite mode].

⁶² Principles II.39 (AT VIIIA 63/CSM I 242). ⁶³ On the question of laws, I believe Curley’s interpretation deserves to be called the ‘orthodox’ position in the literature. Both Steven Nadler (2006) and Michael Della Rocca (2008) mention Curley’s position on laws of nature as a solution to a number of interpretive problems, though both stop short of fully endorsing it. Jonathan Bennett (1996, 73) writes, ‘[W]hat could such features [viz., ‘features of the extended world that it instantiates always and everywhere’] be? The only convincing answer to this that I know of is Curley’s. He says that infinite modes are causal features of the world, and a statement attributing such a mode to the world would be a basic causal law.’ Other proponents of the Curley solution include Jon Miller (2003).

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And yet every finite mode follows from another finite mode: 1p28: ‘Every singular thing, or any thing which is finite and has a determinate existence, can neither exist nor be determined to produce an effect unless it is determined to exist and produce an effect by another cause, which is also finite and has a determinate existence.’ The general facts reading has an elegant solution to this problem. If we assume that Spinoza’s laws are general facts that govern particular facts, we can say at the same time that any individual finite mode follows from the infinite modes, in the way that a particular fact follows from a general one, and that each finite mode is produced by another such mode. Curley’s scheme looks like this: Attribute of extension: the ‘basic’ nomological facts. Immediate infinite mode of extension: derivative but primary nomological facts. Mediate infinite mode: derivative and secondary nomological facts. Finite modes: singular facts.⁶⁴

So on Curley’s view, a finite mode, plus a law of nature, cooperate to produce a new finite mode. Even those who do not explicitly endorse all of Curley’s views are friendly to his reading of laws. Jon Miller, for example, writes, ‘laws are real members of the world and they (along with the series of causes of which all finite modes are members) directly determine all that comes to happen or exist.’⁶⁵ Curley goes further: Spinoza ‘identified God with Nature, not conceived as the totality of things, but conceived as the most general principles of order exemplified by things.’⁶⁶ There are many reservations one might have about this interpretation. If monism just amounts to the claim, ‘there is only one set of general principles,’ it is hard to see why anyone would object. Nor is there much ground for thinking that modes of any kind are facts. Nor is there any compelling textual basis for treating motion and rest, for example, as derivative nomological facts as opposed to motion and rest. The key to finding our way out of the problem posed by 1p23 and 1p28 is to see, with Richard Mason, that the relation between infinite and finite modes ‘was not meant to be a causal one’ in the first place.⁶⁷ Once we stop assuming that the sense in which finite modes follow from the infinite ones has to be causal, we can see our way clear to a reading more grounded in the text. Let’s begin with the immediate infinite mode of extension. Spinoza tells us it is motion-and-rest.⁶⁸ Recall that an immediate infinite mode is ‘the mode which, in ⁶⁴ Adapted from the chart given by Curley (1969, 63). In epistemic terms: ‘[W]e can look on the fundamental laws of nature not only as principles which explain whatever happens in nature, but also as principles which could not, by their very nature, be explained by anything else’ (Curley 1988, 44). ⁶⁵ Miller (2003, 266). ⁶⁶ 1988, 42; italics in original. ⁶⁷ Mason (1986, 204). ⁶⁸ In the Short Treatise, Spinoza claims that the infinite mode (there is no mention of mediate as opposed to immediate modes) in extension is simply matter (1994, 85). If, as I shall argue, the

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order to exist, needs no other mode in the same attribute.’⁶⁹ Given Spinoza’s nominalism, we should not read him as here implying that there is a Platonic universal, motion-and-rest, that emanates from the attribute of extension. Instead, this is a claim about extended things: any extended thing will, solely in virtue of being extended, be either in motion or at rest. What are we to make of the famously obscure mediate infinite mode of extension, facies totius universi? The most natural reading of the phrase, at first glance, takes it to be simply the sum total of every extended ‘thing,’ i.e., the aggregate of all finite modes of extension.⁷⁰ In other words, it is the super-individual of 2p13 and of Letter 32.⁷¹ Now, a mediate infinite mode should depend on nothing but the immediate mode. Putting all this together, we have the claim that the sum total of all finite modes of extension depends on nothing but motion-and-rest. And this is unsurprising, since Spinoza’s bodies are individuated by motion and rest. We need appeal to nothing other than motion-and-rest to explain the face of the entire universe.⁷² So although the finite and infinite modes do not causally cooperate, they do stand in dependence relations, and that is all we need to make sense of 1p23 and 1p28.

5. Conclusion The way of laws is not just crooked but forked. Pursued in one, broadly Cartesian, direction, it leads to occasionalism. For this top-down approach divorces laws from the things that obey them. The immediate question is: what is responsible for the enforcement of the laws? The only candidates for that role in the early modern period are Descartes’s God and Newton’s mysterious ‘agent acting constantly according to laws.’⁷³ I have been trying to unearth a distinct notion of law that retains its legal

(immediate) infinite mode of extension underwrites the production of finite modes, it makes sense that he adds ‘rest’ to the mix. ⁶⁹ As Bennett reminds us (2001, v.1, 171); the quotation is from Spinoza’s Short Treatise (1661) in Spinoza (1985, 153n). ⁷⁰ See Mason (1986, 205–7) for a fuller defense of this point. As Mason acknowledges, Spinoza is on shaky ground when he supposes that a mere aggregate of infinite modes could itself be infinite. ⁷¹ I owe this point to Tad Schmaltz (1997). ⁷² So construed, Spinoza’s point would be analogous to Descartes’s claim in Le Monde that God can, simply by producing motion, create the world as it appears to us (CSM I 91/AT XI 34–5). Spinoza’s point differs from Descartes’s, however, in two key respects: a) Descartes appeals to the laws of nature to explain the order and perfection of the universe, and b) Descartes thinks matter would be arranged in such a way as to ‘have the form of a quite perfect world.’ ⁷³ See the Letter to Bentley of 25 February 1692/3, in Newton (2004). Newton argues that ‘it is inconceivable that inanimate, brute matter should, without the mediation of something else, which is not material, operate upon and affect other matter without mutual contact’ (2004, 102). ‘Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of your readers’ (2004, 103).

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flavor but ties it to objects and their powers.⁷⁴ For Bacon and for Spinoza, nature takes the course it does because it has the powers, that is, the laws, that it does. There is no lawgiver; there is only power. For both Bacon and Spinoza, laws are inscribed in things. They take the very same legal metaphor Descartes does but preserve a different aspect of it. Where Descartes retains the air of arbitrariness the metaphor conveys, claiming that it is God’s nature and not that of creatures that determines the laws, Bacon and Spinoza appeal to laws to bring out the multi-faceted ways in which objects and powers interact. To call a thing’s powers its laws is to emphasize at once its enduring and unchanging capabilities and the conditional status of their exercise. In this respect, Bacon and Spinoza have more in common with their predecessors than their immediate successors. That the notion of laws as powers looks backward rather than forward will be seen by some as a point against my reading.⁷⁵ To this there are two replies. First, the fact that some philosophers should try to defend parts of a worldview that was quickly becoming outmoded is neither surprising nor something to be apologized for. What is more important, while their fellow philosophers in the modern period largely abandon the powers-based ontology, it has experienced a resurgence in the last forty years or so. Indeed, contemporary authors such as Brian Ellis speak of a power’s ‘law of action,’ which ‘describes the kinds of changes that must result when the causal power is activated in circumstances of the appropriate kind.’⁷⁶ Ellis is reproducing, if unconsciously, the very concept of law we have recovered. Seen from this wider historical perspective, it is the Cartesians who are clinging to a doomed raft, not Bacon and Spinoza.⁷⁷

⁷⁴ This use antedates even Francis Bacon. It occurs in Roger Bacon’s Opus Majus of 1267, where we find the claim that the transmission of disease ‘is thus an amazing power, since everything happens in accordance with its laws, both hidden and manifest’ (1267/1928, vol. 1, 163). ⁷⁵ Considering something like the view I defend, Richard Manning (2012) argues that ‘Another, related problem for this interpretation is that it represents Spinoza as, not an avant-garde thinker anticipating modern physics, but as a rear guard defender, despite his official anti-scholastic stance, of the traditional neo-Aristotelian doctrines of essence and substantial form, open to the same charges of ad hoc theorizing and appeal to occult powers that Modernity and the Scientific Revolution leveled against it in their rise to intellectual dominance.’ ⁷⁶ Ellis (2002, 172). ⁷⁷ I would like to thank Galen Barry, Michael LeBuffe, Jim Darcy, Mason Pilcher, and especially Antonia LoLordo for helpful comments. An earlier version was presented as a keynote address to the Rotman Summer Institute, ‘Causal Powers in Science: Blending Historical and Conceptual Perspectives,’ in July 2014. I am very grateful to the conference organizer, Henrik Lagerlund, for the invitation to speak. The paper benefited from many comments and criticisms, especially those of Deborah Brown, Jennifer McKitrick, and Calvin Normore.

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5 Laws and Powers in the Frame of Nature Stathis Psillos

Till a philosopher, at last, arose, who seems, from the happiest reasoning, to have also determined the laws and forces, by which the revolutions of the planets are governed and directed. David Hume (First Enquiry, section I, 14)

1. Introduction Within the Aristotelian framework of natural philosophy, which was dominant until the emergence of the mechanical philosophy, activity, motion, and change in nature were taken to be grounded in powerful substances. From the seventeenth century on, this view started to recede in favour of a conception of nature as governed by general laws of nature and of motion in particular. This new conception was introduced, almost single-handedly, by Rene Descartes, but it spread quickly.¹ Part of the motivation was a thorough critique of the widespread view that, when it comes to natural bodies, they possess active and passive powers in virtue of which they interact with each other. Natural powers were taken to be necessity enforcers in that they were, in and of themselves, principles of necessitation: given the power of X to Φ, X must Φ when the appropriate circumstances arise. Hence, powers were regularity enforcers: they accounted for the regularity there is in the world. The new conception of laws emerged as an alternative to powers. Corporeal substances were widely taken to be passive and inert. Hence, activity, motion, and change could not arise from within a powerful matter. They had to be imposed on matter from without and this meant that there was a need for new principles of connection, viz., principles which determined the ways pieces of matter move and ¹ There is important historical work concerning the origins of the concept of law of nature. For a thorough discussion and support of the claim that Descartes should be credited with initiating the modern conception, see Henry (2004).

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collide with each other. Laws of nature were broadly taken to be the required principles of connection. This very move was itself based on a thorough reconceptualization of the very idea of natural law. The notion was widely applicable to rational beings only, since the dominant thought was that only rational beings can obey the law. After Descartes, the concept of law was widely taken to apply to all beings; and in particular to passive corporeal substances.² The new principles of connection—laws of nature—retained a key feature of the powers-based account of activity in nature, viz., necessity. They were meant to hold by necessity, though what kind of necessity this is was very much in dispute. They were also meant to necessitate the behaviour of things: things had to obey the laws that governed their behaviour. But if there is no power in matter, how does matter act on matter? Broadly speaking, there are two options available, given the passivity of matter. The first is that matter exists but God is the only motive force (this is the line followed by Malebranche and came to be known as Occasionalism). The other is to deny the existence of matter altogether (since there is nothing for it to do) and to claim that God is the direct cause of all ideas in minds (this leads to Berkeley’s idealism). On both options, laws play a key role: they replace powers and provide the missing connections between distinct existences. The aim of this chapter is to revisit the major arguments of the seventeenthcentury debate concerning laws and powers. Its primary points are two. First, though the dominant conception of nature was such that there was no room for power in bodies, the very idea that laws govern the behaviour of (bits of) matter in motion brought with it the following issue, which came under sharp focus in the work of Leibniz: how possibly can passive matter, devoid of power, obey laws? Though Leibniz’s answer was to reintroduce powers, two radically different conceptualizations of the relation between laws and powers became available after him. Hume denied powers altogether, whereas Newton thought that to introduce a power is to introduce a law. The second main point will be that though laws were meant to replace powers, the real dilemma ended up being not laws vs. powers, but rather necessity vs. non-necessity in nature. To exploit an expression used by Newton, the question was: what is the place of necessity in the frame of nature?

2. Against Laws; against Powers For medieval thinkers, natural agency was fundamental and explicable in terms of the powers of the natural agents themselves. God, however, was taken to be the author of the order of the world and its ordering to an end. So, though there is order in the world and in this sense God is the Divine order maker, He does not posit laws of nature over

² Hobbes resisted this trend and refrained from extending natural laws to non-rational beings. For more on this see Goudarouli and Psillos (forthcoming).

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and above the active natures of natural things. Insofar as there is talk of ‘laws of nature,’ this is a metaphorical extension of the concept from the realm of free agents to nature.³ One can see this kind of approach very clearly in the work of the late medieval thinker Francisco Suárez. According to Thomas Aquinas:⁴ ‘Law (lex) is a certain rule and measure in accordance with which one is induced to act or is restrained from acting.’⁵ Thus put, Suárez notes, the definition of law is ‘too broad and general’ precisely because it allows that it is applicable to non-sentient beings, since ‘everything has its own rule and measure, in accordance with which it operates and is induced to act or is restrained therefrom.’ For Suárez, however, laws, strictly speaking, require rational agency. A law is something that has to be (and can be) obeyed (executed) and this can only be satisfied by rational creatures. In this sense, the claim that non-sentient beings obey laws would amount to the claim that they act according to their natural powers. Indeed, insofar as this metaphorical sense of law is used, it can only refer to the orderly action of natural bodies ‘in accordance with the inclinations imparted to them by the Author of Nature.’⁶ Seen that way, talk of natural laws is talk about natural powers. It captures the natural necessity there is in nature and it is to this natural necessity that can be ‘metaphorically given the name of law.’⁷ Note, in addition, that if talk of laws of nature is talk about powers, there are as many ‘laws of nature’ as there are powers—a law for each power in nature. Laws of nature, then, play no role whatsoever as unifiers. Still, if this metaphorical use is made, there must be a law maker who governs the things that ‘obey the law.’ As Suárez put it, insofar as non-rational beings are said to obey a law, they are ‘in need of a superior governing mind . . . and thus from every standpoint, law must be related to mind.’ In sum, then, talk of laws is metaphorical and its proper content concerns natural necessities which are grounded in the natural inclinations of things imparted on them by the law maker. The emergence of mechanical natural philosophy in the seventeenth century brought with it a war against (natural) powers. Qua a sui generis category that explains or grounds motion and change in nature, power came under heavy fire. Here are the main arguments against powers.⁸ The connection problem: how are powers connected? How is it that X having the power to Φ and Y having the power to be Φ-ed (two distinct existences) interact to give rise to an effect? According to Descartes, the power of X to Φ is not sui generis but should be understood in terms of matter in motion. All there is to explain how X acts on Y is action by contact of the parts of X on the parts of Y. In his Le Monde, Descartes objected to the medieval Aristotelian view that fire has the

³ This point has been recently stressed by Marilyn McCord Adams (2013). ⁴ Aquinas (Summa, I–II q. 90 art 1). ⁵ Suárez (1944, 21). ⁶ Suárez (1944, 22). ⁷ Suárez (1944, 22). ⁸ For further insightful discussion, see Ott (2009, chapter 5).

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(sui generis) power to burn wood and argued that all there is to the action of fire on wood can and should be understood in terms of the violent and incessant motion of the parts of fire: When [fire] burns wood or other similar material we can see with our eyes that it moves the small parts of the wood, separating them from one another, thereby transforming the finer parts into fire, air, and smoke, and leaving the larger parts as ashes. Someone else may if he wishes imagine the ‘form’ of fire, the ‘quality’ of heat, and the ‘action’ of burning to be very different things in the wood. For my own part, I am afraid of going astray if I suppose there to be in the wood anything more than what I see must necessarily be there, so I am satisfied to confine myself to conceiving the motion of its parts. For you may posit ‘fire’ and ‘heat’ in the wood, and make it burn as much as you please: but if you do not suppose in addition that some of its parts move about and detach themselves from their neighbours, I cannot imagine it undergoing any alteration or change.⁹

In effect, Descartes’s point is that even if powers were posited, they would fail to explain, in and of themselves, change in nature; the proper explanation would require reference to the action of the parts of the bodies that bear the ‘active power’ on the parts of the bodies that bear the ‘passive power.’ But this very move would make powers redundant, since all the action would be due to the motion of the parts of bodies. This kind of argument against powers is directed primarily against viewing powers as sui generis. Power is not a specific kind of cause of motion/change, distinct for each kind of motion/change. All motion/change in nature should be understood as being of the same kind, viz., the result of action by contact between the parts of matter. Objecting to the view that powers are sui generis principles of motion, Descartes noted: ‘The Philosophers also posit many motions which they believe can occur without any body’s changing place, such as those they call motus ad formam, motus ad calorem, motus ad quantitatem (motion with respect to form, motion with respect to heat, motion with respect to quantity) and countless others.’¹⁰ But he added: ‘I know of no motion other than that which is easier to conceive of than the lines of geometers, by which bodies pass from one place to another and successively occupy all the spaces in between.’¹¹ Hence, all motion should be understood univocally as local motion, and if a sense of power is still operative it can at best be seen as the power of matter to be moved (mobility). For Descartes, then, there are no sui generis powers in nature. To understand all change in nature it is enough to conceive the motions of the parts of matter—subject to universal laws, as we shall see in the next section. Speaking again against the alleged sui generis power of fire to burn, Descartes noted: Once we appreciate that the parts of the flame move in this way, and that to understand how the flame has the power to consume the wood and to burn it, it is enough to conceive of their ⁹ Descartes (1998, 6).

¹⁰ Descartes (1998, 26).

¹¹ Descartes (1998, 27).

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motions, I ask you to consider whether this is not also sufficient for us to understand how the flame provides us with heat and light. For if this is the case, the flame will need possess no other quality, and we shall be able to say that it is this motion alone that is called now ‘heat’ and now ‘light,’ according to the different effects it produces.¹²

The occultness problem: This is a version of the connection problem above. But it is worth treating it separately because the emphasis is shifted to the claim that, in principle, there is no understanding of how a power brings about its effect. In a letter to Regius in January 1642, Descartes noted: But no natural action at all can be explained by these substantial forms, since their defenders admit that they are occult and that they do not understand them themselves. If they say that some action proceeds from a substantial form, it is as if they said that it proceeds from something they do not understand; which explains nothing. So these forms are not to be introduced to explain the causes of natural actions.¹³

We may call this the explanatory impotence argument. Substantial forms were individuating principles that by being joined with specific parcels of prime matter, they made them into substances with a specific causal profile. So, for instance, the substantial form of hard bodies was different from the substantial form of fluid bodies and this primitive difference explained their different powers. But for Descartes no explanation of the difference is thereby achieved: the difference is simply posited as a primitive fact. For him, the only difference between hard bodies and fluid bodies is ‘that the parts of the one can be separated from the whole much more easily than those of the other.’¹⁴ This amounts to a genuine explanation of why there are two distinct kinds of qualities—hardness and fluidity—the difference being grounded in the micro-parts of bodies and their motions. Instead of taking these qualities as primitive, Descartes explains them by showing that they are different species of motion of the particles of matter. Here is how he put it: If you find it strange that, in explaining these elements, I do not use the qualities called ‘heat,’ ‘cold,’ ‘moistness,’ and ‘dryness,’ as the Philosophers do, I shall say that these qualities appear to me to be themselves in need of explanation. Indeed, unless I am mistaken, not only these four qualities but all others as well, including even the forms of inanimate bodies, can be explained without the need to suppose anything in their matter other than motion, size, shape, and arrangement of its parts.¹⁵

The directedness problem: how can a physical quality be directed to another? This was, in many ways, the most puzzling feature of powers. Medieval-Aristotelian powers were endowed with esse-ad. This was supposed to show how ‘one thing receives something from another or confers it upon the other’ (Aquinas) without the qualities of each thing being shared by the other. But the puzzle was precisely how a quality

¹² Descartes (1998, 8). ¹⁵ Descartes (1998, 18).

¹³ Descartes (1984a, 208–9).

¹⁴ Descartes (1998, 10).

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can be inherent to a subject and ‘bear’ on another. For Descartes, this relatedness can only be a feature of minds and their content: only minds can be directed towards anything. Hence, either matter should be endowed with minds or there can be no such thing in matter as ‘being directed to an effect.’ The first horn is precisely the one Descartes dismissed by what came to be known as ‘the little souls argument.’ In a letter to Mersenne (26 April 1643), he noted: I do not suppose there are in nature any real qualities, which are attached to substances, like so many little souls to their bodies, and which are separable from them by divine power. Motion, and all the other modifications of substance which are called qualities, have no greater reality, in my view, than is commonly attributed by philosophers to shape, which they call only a mode and not a real quality.¹⁶

The reference here is to the medieval idea that qualities can exist independently of substances. But what’s interesting is that Descartes takes it that these qualities were endowed with directedness as if they were soul-like entities. As he explained in a letter to Princess Elizabeth, 21 May 1643, real qualities were conceived by means of notions that were used ‘for the purpose of knowing the soul.’¹⁷ Gravity, one of Descartes’s favourite examples, is a case in point. Descartes noted that instead of taking the view that heaviness is a kind of motion which is ‘produced by a real contact between two surfaces,’¹⁸ his medieval predecessors, starting from the ‘inner experience’ of how the soul operates on bodies, took it that heaviness is a real quality ‘of which all we know is that it has the power to move the body that possesses it towards the centre of the earth,’¹⁹ thereby wrongly attributing to it soul-like attributes. Speaking critically of his former self in the sixth set of replies he noted that he too used to ascribe to gravity some kind of directed power to carry bodies towards the centre of the earth ‘as if it had some knowledge of the centre.’ But he changed his mind when he realized that this was tantamount to applying mind-like properties to gravity; for this directedness ‘surely could not happen without knowledge, and there can be no knowledge except in a mind.’²⁰ The motive force problem: How can a body move itself or another body? This question was thrown into sharp relief by Nicolas Malebranche who concluded that the only motive power is (in) God. According to a widely accepted account of God’s action, God continuously annihilates and re-creates everything there is. When God wills a body A to come to existence, he wills it to be in a specific space a at a specific time t. How, then, Malebranche asks, can body A move from point a to another point b all by itself? Or how can it be moved by the ‘power’ of another body B? In either case (self-motion or motion-by-other) the required power would ‘supersede’ the power of God, who willed body A to be in point a. Since nothing can supersede God’s power, only God has the power to move a body. Malebranche concluded that ‘there is a

¹⁶ Descartes (1984a, 216). ¹⁹ Descartes (1984a, 219).

¹⁷ Descartes (1984a, 219). ²⁰ Descartes (1984a, 298).

¹⁸ Descartes (1984a, 219).

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contradiction in saying that one body can move another,’ since ‘no power, however vast it may be imagined to be, can surpass or even equal the power of God’; ‘no power can transport [a body] whither God does not transport it, nor fix or keep it where God does not fix or keep it, if it is God alone who adapts the efficacy of His actions to the ineffective actions of His creations.’²¹ Arguments such as the above cast considerable doubt on the idea that matter is powerful. But if there is no power in matter, what options are available? Laws of nature replaced powers as principles of connection between distinct existences. But laws did not replace necessity. Instead, they captured the necessity that was taken to be grounded in powers. But how do laws acquire their necessity and what kind of necessity is this?²²

3. Cartesian Laws In Descartes’s Principia, which appeared in 1644, God is the primary cause of all motion. But laws of nature become themselves the particular causes ‘by which individual parts of matter acquire movements which they did not previously have.’²³ They account for changes of states of motion of bodies: ‘the rules or laws of nature . . . are the secondary and particular causes of the diverse movements which we notice in individual bodies.’²⁴ But what does that mean? How can laws be secondary causes? To answer this question let us first take a look at Descartes’s three laws of nature. In the Cartesian picture of things, God created matter in motion and rest ‘and now maintains in the sum total of matter, by His normal participation, the same quantity of motion and rest as He placed in it at that time.’²⁵ Throughout his work, this is a fundamental idea: the conservation of the total quantity of motion placed in nature by God initially. This is not a law—it is a governing principle. It follows directly from the immutability of God. In fact, it can be said that this Principle of Conservation of the Quantity of Motion (PCQM) is simply a facet of God.²⁶ Note that God might not have chosen to put matter in motion. Hence, PCQM depends on God’s Will in the sense that He willed to set matter in motion. But given this volition, PCQM is metaphysically necessary. God being immutable, PCQM has to hold in all possible worlds in which matter is in motion. The three fundamental laws of nature are grounded in PCQM, that is in the immutability of God. They are part of the fabric of the world. Given his will to set ²¹ Malebranche (1923, 189). ²² For lack of space I do not discuss Descartes’s account of laws and power in Le Monde. Suffice it to say that it is arguable that Descartes changed his position in moving from Le Monde to his mature work Principia. Briefly put, in Le Monde laws play a causal role, but matter is causally active too. For further discussion, and for what I take it to be the right account of the transition from Le Monde to Principia, see Ott (2009, 55ff). ²³ Descartes (1984b, 58). ²⁴ Descartes (1984b, 59). ²⁵ Descartes (1984b, 58). ²⁶ For some similar thoughts see Garber (2013).

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matter in motion, PCQM and the fundamental laws are a constitutive part of nature: they follow from his immutability. Everything that happens in nature is subject to them. Motion, in other words, is regulated by God via laws, subject to an overall PCQM. Descartes says: ‘because God moved the parts of matter in diverse ways when He first created them, and still maintains all this matter exactly as it was at its creation, and subject to the same law as at that time; He also always maintains in it an equal quantity of motion.’²⁷ Then come the three laws. The first law: ‘that each thing, as far as is in its power,²⁸ always remains in the same state; and that consequently, when it is once moved, it always continues to move.’²⁹ For Descartes motion and rest (lack of motion) are two distinct states of matter. Rest (change of state of motion to Rest) requires external agency (collision). Nothing moves from being in motion to being in lack of motion (Rest) ‘by virtue of its own nature’ because this would mean that it would move ‘toward its opposite or its own destruction.’ This is the justification of the first law. Descartes notes that this law ‘results from the immutability and simplicity of the operation by which God maintains movement in matter.’³⁰ What’s important to add here is that Descartes explicitly denies that bodies are endowed with tendencies or capacities to change their state of motion ‘by virtue of their own nature.’³¹ The second law: ‘that all movement is, of itself, along straight lines; and consequently, bodies which are moving in a circle always tend to move away from the center of the circle which they are describing.’³² This law captures the idea that circular motion is ‘forced’ or ‘constrained motion’: it is deviation from the straight path due to the encounter of other bodies or due to the rigid connections among the parts of a body (as in the motion of a wheel). Descartes’s justification is based directly on the immutability of God and the simplicity of his operations. This law too, he says, ‘like the preceding one, results from the immutability and simplicity of the operation by which God maintains movement in matter.’³³ The appeal to the immutability of God is essential because God maintains motion ‘precisely as it is at the very moment at which He is maintaining it, and not as it may perhaps have been at some earlier time.’³⁴ But the motion at each instant (or moment) is along a straight line. Though ‘no movement is accomplished in an instant,’ Descartes considers it obvious that ‘every moving body, at any given moment in the course of its movement, is inclined to continue that movement in some direction in a straight line, and never in a curved one.’³⁵ Simplicity, however, ²⁷ Descartes (1984b, 58). ²⁸ The Latin expression is quantum in se est. Though it is usually rendered as ‘as far as is in its power,’ it is better rendered as ‘as far as it is [in itself]’. ²⁹ Descartes (1984b, 59). ³⁰ Descartes (1984b, 60). ³¹ Descartes (1984b, 59). ³² Descartes (1984b, 60). ³³ Descartes (1984b, 60). ³⁴ Descartes (1984b, 60). ³⁵ Descartes (1984b, 60).

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is important, because motion along a straight line is simpler than curvilinear motion. From this law, Descartes says, it follows that ‘any body which is moving in a circle constantly tends to move [directly] away from the center of the circle which it is describing.’ Hence, the centrifugal tendency of a rotating stone in the sling is grounded in the second law. The third law: ‘that a body, upon coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body.’³⁶ Descartes proves the third law by appealing to the immutability of God. In the creation, God gave motion to particles and caused some of them to collide with others. Because he is immutable, he conserves motion in particles; hence the total quantity of motion, including that which is transferred from one particle to another, remains the same: ‘in now maintaining the world by the same action and with the same laws with which He created it, He conserves motion; not always contained in the same parts of matter, but transferred from some parts to others depending on the ways in which they come in contact.’³⁷ How then are laws the ‘secondary and particular causes of the diverse movements which we notice in individual bodies’?³⁸ Given that bodies were set in various states of motion in the beginning, laws cause (i.e., determine) their subsequent states. Why does X move in a straight line? Because it is a law that it must keep moving (it won’t change its state of motion) until something stops it (First law). In this sense, the law is a cause of the movement of X, since it dictates (determines) its motion. Similarly, why does X stop moving? (Or why does X change its state of motion?) Because it is a law that X must change its state of motion if it collides with another body Y. Hence, change of the state of motion of X is a law-governed collision with another body Y (Third law). Finally, why does X move in a curved path? Not because ‘it is inclined to any circular movement’; but because it is a law that motion along curved path AB is forced motion, that is it consists in a deviation from the straight line (Second law). It is striking that in the Principia, there is no reference to the dispositions of matter as causes of the motion of the bodies alongside the laws of nature.³⁹ To be sure, Descartes does talk about affections, but he states clearly that all dispositions [omnium affectionum] of matter arise out of the movements of its parts; hence ³⁶ Descartes (1984b, 61). ³⁷ Descartes (1984b, 62). ³⁸ Descartes (1984b, 59). ³⁹ In Le Monde Descartes notes that though when it comes to rectilinear motion, it is fully determined by the first two laws and God’s conservation of things the way they are, in the case of circular motion, an extra explanatory principle is required, viz., the dispositions of matter. This explains why though the parts of a body tend to move in a straight line, the body as a whole moves in a circle. Here is how he put it: ‘According to this rule [the third law of motion], then, we must say that God alone is the author of all the motions in the world in so far as they exist and in so far as they are straight, but that it is the various dispositions of matter that render the motions irregular and curved. Likewise, the theologians teach us that God is also the author of all our actions, in so far as they exist and in so far as they have some goodness, but that it is the various dispositions of our wills that can render them evil’ (1998, 30).

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there are no sui generis (non-mechanical) dispositions.⁴⁰ He explicitly states that when he uses expressions such as ‘striving [conatus] of inanimate objects toward motion,’ he does not attribute any thought or intention to them; he simply means ‘that they are so situated, and so urged to move [ad motum incitatos], that they will in fact recede if they are not restrained by any other cause.’⁴¹ And in IV.199, he sums up his view by saying that all natural phenomena are ‘nothing other than, certain dispositions [dispositiones] of size, figure, and motion {of bodies}.’⁴² So there is no reason to think that for Descartes the references to dispositions, affections, and the like in the Principia are anything other than the (passive) mechanical qualities of the corpuscles of matter. In fact in Principia, forces and generic powers of matter are fully replaced by laws.⁴³ In a characteristic passage in II.43, Descartes states that ‘the force of each body to drive or to resist consists’ in the body’s obeying the first law of motion—period.⁴⁴ Descartes says that the ‘force of each body to act against another or to resist the action of that other consists . . . in the single fact that each thing strives, as far as is in its power [quantum in se est], to remain in the same state, in accordance with the first law stated above.’⁴⁵ Forces of action or resistance are replaced by law-like behaviour. Here is a summary of the main features of Cartesian laws. Laws are difference-makers: all phenomena in nature are counterfactually dependent on laws. If the laws had been different (per impossibile), the world would have been different. Laws are counterfactually robust: No matter what the initial arrangement of particles in motion, the laws are such that the effect would be the same. Which effect? The world as we know it. Laws are metaphysically necessary: God agitated the different parts of matter in diverse ways, but then He ‘did no more than sustain nature in His usual manner leaving it to act according to the laws He has established.’ These laws are such that ‘even if God had created many worlds, there could be not be any in which they could have failed to be observed.’⁴⁶ Laws are immanent: they directly follow from God’s immutability. Laws are causal principles: the motion of the various bodies in the world happen because of them and not simply according to them.

⁴⁰ Cf. Descartes (1984b, 50). ⁴¹ Descartes (1984b, 112). ⁴² Descartes (1984b, 282). ⁴³ Here I am in agreement with Garber (1992, 298) who takes the line that Cartesian forces are nothing but ways of talking about how God acts on bodies in a lawful way. But, as it will become clear later on, I add to Garber’s account that, strictly speaking, Descartes replaces forces with laws in the sense that there is nothing more to the content of force talk than whatever is involved in the law-governed motion of bodies. For more on the various accounts of force in Descartes see Garber (1992, chapter 9), Ayers (1996), and Ott (2009, chapter 8). ⁴⁴ Descartes (1984b, 63). ⁴⁵ Descartes (1984b, 63). ⁴⁶ Descartes (1984a, 132).

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The passivity of matter, however, created a potential tension in the Cartesian system: How are laws executed by matter? If matter is inert and lacks any motor force, how is it that pieces of matter are subjected to the causal action of laws? Descartes is clear that bodies have ‘no innate tendency to motion.’⁴⁷ Nor do they have any ‘tendency towards rest.’⁴⁸ All action is by contact and requires a mover. He therefore puts forward seven rules by virtue of which the collision of bodies is regulated. Though these rules have been found wanting, the relevant point here is that Descartes introduces them as rules which ‘determine to what extent the movement of each body is changed by coming in contact with other bodies.’⁴⁹ This way to formulate the task is fully consistent with the claim that matter is passive and hence that rules (laws) determine how (the quantity of) motion is redistributed among the colliding bodies. But in showing how these rules apply to contacts among moving bodies Descartes stated that ‘it is only necessary to calculate how much force to move or to resist movement there is in each body; and to accept as a certainty that the one which is the stronger will always produce its effect.’⁵⁰ Hence, it is not clear, to say the least, that Descartes has a coherent view about how matter is capable to act on matter and be acted upon by matter. In a letter to More (August 1649), Descartes noted that a created substance can have the power to move a body; but this is a mode of the created substance conferred on it by God.⁵¹ Still, can motion be transferred from one body to another? If motion is a mode of the thing moved (as it certainly is), then it cannot since no mode can be transferred from one body to another. In Principia Descartes defined motion as the translation of one part of matter or of one body, from the vicinity of those bodies immediately contiguous to it and considered as at rest, into the vicinity of [some] others . . . I also say that it is a translation, not the force or action which transfers [non vim vel actionem quae transfert], in order to show that this motion is always in the moving body and not in the thing which moves it (because it is not usual to distinguish between these two with sufficient care); and in order to show that it is only a mode [of the moving body], and not a substance, just as shape is a mode of the thing shaped, and rest, of the thing which is at rest.⁵²

In this critical passage, he distinguished clearly between motion as translation (i.e., change of relative place) and motion as force or action which is transferred from one body to another. But in many other places in the Principia Descartes talks freely of the ‘transference’ of motion.⁵³ A charitable reading of Descartes would be this. Strictly speaking, there is no transfer of motion between bodies (though, loosely speaking their collisions can be described as if there were transfer). Still, when bodies collide with each other, the quantities of motion they already possess are re-distributed among them according to the relevant laws (and always in line with

⁴⁷ Descartes (1984b, 94). ⁵⁰ Descartes (1984b, 64). ⁵³ See e.g., Part II, 40 and 42.

⁴⁸ Descartes (1984b, 59). ⁵¹ Cf. Descartes (1984a, 381).

⁴⁹ Descartes (1984b, 64). ⁵² Descartes (1984b, 51).

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the overarching principle PCQM). The laws then are the (particular and secondary) causes (difference makers) of the redistribution of particular quantities of motion, since without them there would be no fact of the matter as to what would happen in the collision.

4. Occasionalist Laws The occasionalist reaction to Descartes was based on the thought that if matter is truly inert and lacking of any motive power, then the only real (and direct) cause should be God himself. For Malebranche, we have no conception of power of things. If the only motive power is (in) God, how does God act on matter? The answer is clear: via laws. God acts via laws because his action is uniform and simple; it ‘link(s) together the parts which compose the world’ and makes the world knowable.⁵⁴ Hence, laws are willed by God. Laws are principles by means of which God acts in nature and establishes its uniformity—which is a facet of its perfection. For Malebranche, God submits to laws not by absolute necessity but by desire.⁵⁵ Being omnipotent, God could have created an infinity of possible worlds. Hence, there could have been a law-less (disorderly) world. But God did not desire to create such a world; hence, he did not will it. God created an orderly world, governed by the simplest laws since He acts ‘always by the simplest ways.’⁵⁶ In fact, of all the possible (law-governed) worlds, he willed to create ‘that world which could have been produced and preserved by the simplest laws, and which ought to be the most perfect with respect to the simplicity of the ways necessary to its production or to its conservation.’⁵⁷ So perfection is a function of the simplicity of the ways the world could have been created and conserved. As he says later on: ‘if he had been able (by equally simple ways) to make and to preserve a more perfect world, he would never have established those laws.’⁵⁸ Unlike Cartesian laws, Malebranchean laws are not grounded in the immutability of God but in his Will and, in particular, in his ways of acting. He always acts in the simplest ways and He always acts (in the natural order) with general laws (volitions). God wills certain laws ‘because of their fruitfulness.’⁵⁹ What then are the laws of nature? They are God’s general volitions. They are his decrees. As such, the laws of nature are not metaphysically necessary. The decrees (laws) of God are only ex hypothesi necessary. Hence, the question ‘Can God change the laws?’ admits of a positive answer. Like Descartes, Malebranche thought that God could not have set matter in motion (if he had wished not to produce anything new in the world). But in addition, God might will that the actual laws of motion change (if, for instance, he wills to create incorruptible bodies). ⁵⁴ Malebranche (1923, 190). ⁵⁷ Malebranche (1992b, 116). ⁵⁹ Malebranche (1992b, 119).

⁵⁵ Malebranche (1992b, 119). ⁵⁸ Malebranche (1992b, 120).

⁵⁶ Malebranche (1992b, 118).

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Though laws could be different (or could change, if God willed it), they are constant and immutable and they hold everywhere and everywhen. He couldn’t be more explicit: ‘The laws of nature are constant and immutable; they are general for all times and for all places.’⁶⁰ In fact, it is the very constancy and immutability of laws that explains why in nature there are unwanted (or undesirable) effects—e.g. hail destroys a harvest or a malformed baby is born. For Malebranche, it is not that God willed, with a particular volition, these particular effects to happen. Rather, these effects are the ‘necessary consequences’ of the laws of communication of motion he had initially established.⁶¹ God desires that ‘all of his creatures should be perfect’;⁶² But he acts by general laws which he foresees to be the most fecund.⁶³ These are also the simplest laws. Precisely because God acts in the simplest ways, it would be ‘unworthy of his wisdom to multiply his wills in order to stop certain particular disorders.’ Hence, simple and general laws ‘cover’ everything that happens in nature, within the natural (i.e., not miraculous) order. The two fundamental laws of nature are laws of motion: The First Law: ‘that moved bodies tend to continue their motion in a straight line’;⁶⁴ The Second Law: ‘that when two bodies collide, their motion is distributed both in proportion to their size, such that they must afterwards move at an equal speed.’⁶⁵ According to the second law, what happens in collisions is the redistribution of motion ‘in proportion to their size.’ As Malebranche noted this law is not observed in experience; but he noted that it is a true law holding in the ‘invisible’ bodies.⁶⁶ In Dialogues on Metaphysics,⁶⁷ the first law of motion was justified on the grounds that the straight line is the simplest and shortest line. When it comes to the second law, he noted that though there is change of direction, there is conservation of the ‘quantity of the moving force.’⁶⁸ These two principles constitute ‘the general laws of the communication of movements in accordance with which God acts incessantly.’⁶⁹ Malebranchean laws of nature are ‘efficacious’;⁷⁰ in particular, it is because of the efficacy of the second law (governing impact) that diversity in matter is produced: the diversity there is in matter is counterfactually dependent on the second law of motion. But more than this, both laws are the causes of all motion: ‘These two laws are the cause of all the motions which cause that variety of forms which we admire in nature.’⁷¹ They are ‘necessary to the production and the preservation of the earth, and

⁶⁰ ⁶² ⁶⁴ ⁶⁶ ⁶⁸ ⁶⁹ ⁷⁰ ⁷¹

Malebranche (1992b, 118). ⁶¹ Malebranche (1992b, 118). Malebranche (1992b, 118). ⁶³ Cf. Malebranche (1992b, 119). Malebranche (1992b, 117). ⁶⁵ Malebranche (1992b, 117). Malebranche (1992b, 117). ⁶⁷ Malebranche (1923, Dialogue 7, XI). Malebranche (1923, 191). Malebranche (1923, 191). Cf. also Malebranche (1997, 664, Researche, Elucidation XV). As Robert Merrihew Adams (2013, 75) aptly put it, Malebranchean laws ‘have ‘oomph.’ Malebranche (1992b, 115).

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of all the stars that are in the heavens.’⁷² Whatever happens in nature (barring God’s miraculous intervention) is the necessary consequence of the laws of nature. But how are laws executed by matter? For Malebranche, they are not! God himself executes the laws. The laws are the divine decrees themselves, which ground the regularity. The laws are principles of connection (‘indissoluble bonds of union’) between natural things (distinct existences). As he put it in Dialogue 7, XIII: ‘the divine decrees are the indissoluble bonds of union between the various parts of the universe and of the marvelous network of all the subordinate causes.’⁷³ As Malebranche stressed, the two basic principles of what came to be known as occasionalism are these: 1. Bodies lack motor force. 2. God acts on nature via general laws. Here is how he put it: ‘these two principles, of which I am convinced, that none but the Creator of bodies can be their mover, and that God communicates His power to us only through the establishment of certain general laws, the realisation of which we determine through our various modifications.’⁷⁴ Since matter is impotent to execute the laws, they must be ‘executed’ by God; yet, as is explicitly stated in the foregoing passage, the laws are ‘realised’ in various natural (i.e., occasional) causes (via their modifications). In this scheme, there is no room for connecting entities like powers/ active qualities. The link between God (as the only power) and the regularity there is in nature is laws—that is God’s decrees which are directly realized in natural bodies and their motions.⁷⁵ Hence, laws being the ‘indissoluble bonds of union between the various parts of the universe’ are executed by God and are realized in natural bodies and their regular behaviour. How are laws executed by God? When He re-creates a body in motion, He makes it move always in a straight line according to the first law, i.e., the subsequent re-creations should be in accordance to the first law. This is God’s general volition. But there are collisions. Again, the law of collisions determines that the re-created bodies will be such that there is a certain redistribution of motion among them. In a collision between body X and body Y, X is not the cause of the motion of Y. X communicates ‘nothing of its own’ to Y. Yet, there is a law-governed redistribution of motion between X and Y. God acts according to a law which causes the redistribution of motion. Hence though all communications of motion are executed by God, they are executed by a general law. The action of natural causes consists ‘only in the motor force activating them,’ i.e., God.⁷⁶ Does that mean that God communicates his power to bodies? Not quite! God has made the modifications of bodies the occasional causes of his action—which is law-governed.⁷⁷ ⁷² Malebranche (1992b, 115). ⁷⁴ Malebranche (1923, 196). ⁷⁶ Malebranche (1997, 662).

⁷³ Malebranche (1923, 195). ⁷⁵ Cf. Malebranche (1923, 195, Dialogues 7, XIII). ⁷⁷ Cf. Malebranche (1997, 225).

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There is an ongoing debate among Malebranche scholars about the content of laws of nature—are they general volitions or (sums of) particular volitions?⁷⁸ As Malebranche explains, when God acts with a particular volition, he acts without the occasional cause, e.g., when I feel pain without being pricked by a pin. Or when a body ‘begins to move itself without being struck by another, or without any change in the will of minds, or in any other creature that determines the efficacy of some general laws, I say then that God moves this body by a particular will.’⁷⁹ But what happens when a ball strikes another? How does God act? Malebranche’s answer is that God moves the second ball by a general volition (a general law). The ball is moved ‘in consequence of the general and efficacious laws of the communication of motion.’⁸⁰ So it seems correct to say that Malebranchean laws are general volitions. Malebranchean laws of nature are not metaphysically necessary. But are they contingent? They are not. As we have seen, God submits himself to laws not out of absolute necessity but out of his will for the good. Malebranche also stresses that, to put it in modern jargon, regularity does not imply causation; necessity is also needed. Hence, ‘Only in the wisdom of God do we see eternal, immutable, and necessary truths. Nowhere else but in this wisdom do we see the order that God Himself is constrained to follow.’⁸¹ In a famous passage in his De la recherche de la vérité, Malebranche makes clear that we can conceive of absolute (metaphysical) necessity only between the will of God and something happening because of this: A true cause as I understand it is one such that the mind perceives a necessary connection between it and its effect. Now the mind perceives a necessary connection between the will of an infinite being and its effect. Therefore, it is only God who is the true cause and who truly has the power to move bodies.⁸²

Since there is no perception by the mind of necessary connections between natural (or occasional) causes and natural (or occasional) effects, these are not real causes. For any natural cause, it is conceivable without contradiction that it can occur without its effect. And of course, God can make it happen that it does occur without its (natural) effect. Only in the case of God’s will is it impossible for us to conceive of it without at the same time conceiving whatever He wills to happen. But, barring miraculous interventions, natural causes invariably precede their natural effects in virtue of God’s general volitions, aka laws of nature. Hence, the regularity there is in nature (in virtue of which it is predictable and knowable) is grounded in (in the sense of being the effect of) the laws of nature. The law is not the regularity itself, but it is the principle (God’s general volition) which grounds the regularity. The occasional cause does not act—though it might seem to us that it does. It is the law that is causally efficacious: ‘A body moves immediately after having been ⁷⁸ See Pessin (2001); Ott (2009, chapter 11); Nadler (2011); Robert Merrihew Adams (2013) for a representative sample. ⁷⁹ Malebranche (1992b, 195). ⁸⁰ Cf. also Malebranche (1992b, 195, Illustration 1). ⁸¹ Malebranche (1997, 615–16). ⁸² Malebranche (1997, 450).

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struck: the collision of the bodies is the action of the occasional cause; thus this body moves by a general will.’⁸³ Occasional causes and the concomitant regularity is a sign that the effect does not have ‘something singular about it.’ Hence, the presence of an occasional cause is a sign (‘mark’) that there is action by the general volition of God.⁸⁴ Hence, regularity (occasional ‘causation’) is a ‘mark’ for the presence of a law by means of which God acts. Being the decrees of God, laws hold with some kind of necessity. Recall the very title of Malebranche’s Treatise: On the Necessity of General Laws of Nature. What kind of necessity is this? It is natural or hypothetical necessity. Even though the natural necessity of the laws cannot possibly be rationally inferred from the observation of the invariable conjunction of natural causes and natural effects, it can be conceived if we reflect on the way God acts. The laws of nature are certainly part of the fabric of nature; hence, there is necessity in nature; though this necessity is not located or grounded in bodies ‘by themselves’; nor can it be inferred by experience and from the regularity there is in nature.⁸⁵

5. Berkeley on Laws That laws are naturally necessary is a view that, perhaps surprisingly, can be attributed to Berkeley, too. For Berkeley, all causation is an action of the will of a spirit. Hence, only God and minds (spirits) can be (efficient) causes. Ideas are passive and inert.⁸⁶ Though it is false to claim that ideas are the ‘effects of powers resulting from the configuration, number, motion, and size of corpuscles,’ ideas must nonetheless have a cause ‘whereon they depend, and which produces and changes them.’⁸⁷ These can only be incorporeal substances (i.e., minds qua active substances). God excites (and causes) ideas in us by means of laws of nature. This claim is grounded in the fact that ideas have ‘admirable connexions’ and regularity; they come ‘in a regular train or series.’⁸⁸ So the order and regularity in the co-occurrence and succession of ideas suggest that they are caused in us by a Will which acts in an orderly and regular way. As Berkeley put it: ‘The set of rules or established methods, wherein the mind we depend on excites in us the ideas of sense, are called the “laws of nature”; and these we learn by experience, which teaches us that such and such ideas are attended with such and such other ideas in the ordinary course of things.’⁸⁹ Laws of nature stem from God’s Will. They are expressions of God’s Will to act in a regular way: ‘(T)his consistent uniform working, which so evidently displays the goodness and wisdom of that governing spirit whose will constitutes the laws ⁸³ Malebranche (1997, 450). ⁸⁴ Malebranche (1997, Illustration VIII). ⁸⁵ Here I am in essential agreement with Robert Adams who stresses that for Malebranche there is necessitation in causation in the following sense: occasional causes are ‘part of how God’s general volitions, as efficacious laws of nature, necessitate particular effects’ (2013, 77). ⁸⁶ Berkeley (2008, 92). ⁸⁷ Berkeley (2008, 92). ⁸⁸ Berkeley (2008, §30). ⁸⁹ Berkeley (2008, 94).

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of nature.’⁹⁰ But unlike all of his predecessors, Berkeley took it that laws of nature are known by observation and empirical study. They are not known a priori. They are not inferred by reference to the simplicity of God’s actions or his immutability. They do reflect his goodness and wisdom in creating the world, but they can only be known a posteriori. In particular, laws are not discovered a priori by looking for ‘necessary connections among ideas.’ There is nothing in the idea of fire that necessarily implies that it warms us. The regularity there is in the world is discovered empirically ‘only by the observation of the settled laws of nature.’⁹¹ Thanks to the laws of nature, ideas are connected to each other (e.g., the idea of fire and the idea that it warms us) even though one may be conceived without the other without falling into contradiction. In a certain sense, Berkeley pushed the case for occasionalism to its extremes. Malebranchean occasionalism kept matter but divested it from any causal power or efficacy. Hence, there are no corporeal causes. Without naming it, Berkeley (Principles §53) noted that occasionalism got it right in claiming that ‘amongst all the objects of sense, there was none which had any power or activity included in it, and that by consequence this was likewise true of whatever bodies they supposed to exist without the mind.’⁹² But he thought that the further supposition, viz., that there is ‘an innumerable multitude of created beings, which . . . are not capable of producing any one effect in nature’ is ‘unaccountable and extravagant,’ though possible. If Berkeley found in occasionalism no argument for the philosophical category of matter, he found in it an understanding of how God acts in the world: via laws of nature. In this sense, laws of nature are key to Berkeley’s natural philosophy. As he put it: There are certain general laws that run through the whole chain of natural effects. These are learned by the observation and study of nature, and are by men applied as well to the framing artificial things for the use and ornament of life as to the explaining the various phenomena: which explication consists only in showing the conformity any particular phenomenon has to the general laws of nature or, which is the same thing, in discovering the uniformity there is in the production of natural effects, as will be evident to whoever shall attend to the several instances, wherein philosophers pretend to account for appearances.⁹³

This rich passage suggests the following about laws: a) Laws ‘run through’ the whole chain of natural effects. Hence, laws cover all natural effects (the whole chain of events in nature). This covering should not be seen as ‘governing’ but as ‘running through’—laws ‘permeate’ natural phenomena; they imbue natural phenomena. b) These laws, as noted already, are discovered a posteriori; they are learned by observation and empirical study of nature. ⁹⁰ Berkeley (2008, 94). ⁹³ Berkeley (2008, 107).

⁹¹ Berkeley (2008, 94).

⁹² Berkeley (2008, 103).

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c) Explanation in natural philosophy is showing that particular phenomena conform to a general law; that is, that they are ‘permeated’ by a law. Hence, explanation can be seen as nomological subsumption. d) The ‘production’ of natural effects is nothing more than their law-based explanation—that is, their permeation by laws. God acts via laws at all levels. And because of this, he produces (literally this time, since He’s an efficient cause) any effect ‘according to the standing mechanical laws of nature.’⁹⁴ As Berkeley repeatedly stresses, this is not a necessary truth—it is not ‘absolutely necessary’ for God to produce any effect by mechanical principles. Yet, the metaphysical contingency of the ‘clockwork of nature’ does not detract from the fact that God wills to act ‘agreeably to the rules of mechanism.’ In fact, given that Berkeley does want to accommodate mechanism (the clockwork of nature) within his philosophy, he further argues that although God could produce anything he wanted without any mechanism, the ‘clockwork of nature’ is the way God has chosen to produce effects in nature in a regular and orderly way. Hence, the mechanical laws of nature are conditionally or naturally necessary, viz. necessary ‘to the producing [of an effect] according to the standing mechanical laws of nature.’ In De Motu, in which Berkeley makes an attempt to explain his natural philosophy without explicitly denying that there is matter, he expresses his firm view that ‘(R)egarding body we may boldly declare as established fact that it is not the principle of motion.’⁹⁵ He adds that what we know about the body (what is contained in the idea of body—‘extension, solidity, and figure’) is not a principle of motion. And in §24 he dismisses that there might be something unknown in body which is this principle of motion, because we have no idea of it. Laws of nature, then, ‘replace’ internal principles of motion of bodies. In science it is enough to state true theorems about the motion of the bodies, irrespective of what might or might not cause these motions. These true theorems are ‘the rules and laws of motion’ and the ‘theorems deduced from them’ (regulis & legibus motuum, simul ac theoremata inde deducta). And these laws ‘remain unshaken, so long as sensible effects and reasoning based on them are granted.’⁹⁶ There is little doubt, however, that even though laws are discoverable a posteriori, they hold with natural necessity and permeate the actual behaviour of things.

6. Leibniz on Laws and Powers Malebranche did adopt the principle of the conservation of total quantity of motion. In Dialogue X, he noted: ‘In a word, God has chosen the simplest law on the basis of the unique principle that the stronger shall conquer the weaker; and, subject to this

⁹⁴ Berkeley (2008, 107).

⁹⁵ Berkeley (2008, 250).

⁹⁶ Berkeley (2008, 252).

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condition, that there shall always be in the world the same quantity of motion.’⁹⁷ But he never followed the Cartesian course of trying to ground the laws of motion on this principle and the immutability of God. He did attribute ‘moving force’ to God and claimed that God acts ‘always with the same efficacy or the same quantity of moving force.’⁹⁸ The total quantity of moving force is conserved, since as Malebranche put it, God ‘never changes the quantity of the moving force which animates matter.’⁹⁹ As is well known, a principled disagreement between the Cartesians and Leibniz concerned what exactly is conserved during impact. Already in 1686, Leibniz noted that there is a great difference between the concept of the quantity of motion and the concept of quantity of motive force and that Descartes was mistaken in holding them to be equivalent. This mistake led Descartes, Leibniz argued, ‘to assert that God conserves the same quantity of motion in the world.’¹⁰⁰ But this basic Cartesian principle PCQM was not correct, as Leibniz conclusively demonstrated. What is conserved is the total quantity of vis viva, which is the product of mass times the square of the velocity of a body. As Leibniz put it, what is conserved is the ‘total force and the total direction.’ We will not go into the details of this presently. What is important for our purposes is that Leibniz presents this principle—the conservation of the quantity of force—as a subordinate law of nature.¹⁰¹ The fundamental law of nature is a Law of Order. The world is orderly and regular and for Leibniz ‘no matter how God might have created the world, it would always have been regular and in a certain general Order.’¹⁰² Hence, this fundamental Law of Order is metaphysically necessary. But the most orderly world could have been the most complex one. Leibniz notes that God chose to create the most perfect world, where perfection is a function of two factors—simplicity and strength. As he put it: ‘But God has chosen that world which is the most perfect, that is to say, which is at the same time the simplest in its hypotheses and the richest in phenomena.’¹⁰³ So the simplest world is at the same time the most comprehensive world. This world is structured by subordinate laws. These are God’s general decrees. They govern everything without exception: ‘For the most general of God’s laws, which rules the whole sequence of the universe, is without exception.’¹⁰⁴ The law of the conservation of the quantity of force is the chief subordinate law of nature. It is grounded directly in God in that it is God who ‘always conserves by rule the same force.’¹⁰⁵ Elsewhere, he calls the Law of conservation of quantity of force (PCQF) ‘the foundation of the laws of nature,’ implying that among the subordinate laws there is a hierarchy of laws, with PCQF being at the bottom.¹⁰⁶ But what is force or active power? Leibniz takes it to be a non-mechanical quality, which is necessary for explaining the behaviour of things. It is non-mechanical in the

⁹⁷ ¹⁰⁰ ¹⁰² ¹⁰⁵

Malebranche (1923, 267). Leibniz (1989b, 296). Leibniz (1989b, 306). Leibniz (1989b, 314).

⁹⁸ Malebranche (1923, 191). ⁹⁹ Malebranche (1923, 191). ¹⁰¹ Leibniz (1989b, Discourse on Metaphysics, §17). ¹⁰³ Leibniz (1989b, 306). ¹⁰⁴ Leibniz (1989b, 307). ¹⁰⁶ Leibniz (1989b, 499).

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sense that it is not one of the basic mechanical categories of Cartesianism, viz., extension, figure, and motion. It’s not geometrical and it is attributed to natural bodies over and above their mechanical affections. In fact, given that the basic law of nature is the conservation of the quantity of force, Leibniz argues that in finding the ‘true laws of nature and the rules of motion,’ we have to go beyond mechanics and physics to metaphysics.¹⁰⁷ In physics, we have to ascertain the ‘derivative’ forces (that is the ‘immediate causes’ of motions and changes in nature). Forces are ‘more real’ than their effects (changes in motion) and because forces are ‘different from size, figure, and motion,’ we can conclude that ‘not everything which is conceived in a body consists solely in extension and its modifications.’¹⁰⁸ Leibniz took the need to appeal to forces to vindicate the medieval-Aristotelian view that bodies have active and passive powers, which are not mechanically explicable and grounded. Rather, they ‘pertain to certain forms or indivisible natures’ of corporeal substances.¹⁰⁹ The key characteristic of forces, which are directly ‘implanted’ to corporeal bodies ‘by the Creator,’ is that by being subjected to forces, bodies are endowed ‘with conatus, attaining [their] full effect unless [they are] impeded by a contrary conatus.’¹¹⁰ This force is in the ‘bodies themselves’ and ‘it constitute(s) the inmost nature of the body, since it is the character of substance to act.’¹¹¹ Restoring forces or powers as being inherent in bodies was an important break with Cartesianism. Matter is not passive but active. But Leibniz adopted the idea that the motion of pieces of matter is subject to natural laws, which are established by God. Though all laws other than the fundamental Law of Order are not metaphysically necessary, they are far from ‘arbitrary.’ God has chosen them, but as Leibniz put it, ‘God has been led to set in motion the laws which are observed in nature through determined principles of wisdom and by reasons of order.’¹¹² In this sense, we can say that laws of nature are conditionally or naturally necessary in that given God’s demand for simplicity and strength (i.e., perfection) in an orderly world, these particular laws had to be chosen. In a letter to Malebranche (22 June/2 July 1679), Leibniz puts this point thus: ‘We must also say that God makes the maximum of things he can, and what obliges him to seek simple laws is precisely the necessity to find place for as many things as can be put together; if he made use of other laws, it would be like trying to make a building with round stones, which make us lose more space than they occupy.’¹¹³ His general reaction to occasionalism reveals his views about the relation between law and power. His chief point is that though occasionalists are right in denying that there is interaction among bodies, and in particular, in denying the direct influx theory according to which a cause is what flows into the effect (hence that something flows between two distinct substances), they are wrong in placing all action in God, ¹⁰⁷ Leibniz (1989b, 315). ¹¹⁰ Leibniz (1989b, 440). ¹¹³ Leibniz (1989b, 211).

¹⁰⁸ Leibniz (1989b, 315). ¹¹¹ Leibniz (1989b, 435).

¹⁰⁹ Leibniz (1989b, 315). ¹¹² Leibniz (1989b, 500).

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thereby turning him into a Deus ex Machina. Occasionalism, for Leibniz, introduces ‘a kind of continuous miracle.’¹¹⁴ It is ‘as if God as a rule interfered in some other way than by preserving each substance in its course and in the laws established for it.’¹¹⁵ Though his diagnosis of how God acts according to occasionalism might well be wrong, since for Malebranche at least God need not act by anything other than his general volition, Leibniz points to a genuine problem with occasionalism, viz., that laws are executed directly by God himself and not by natural bodies. So Leibniz took it that a complete understanding of the workings of nature requires both laws of nature and powerful substances. He identified a key problem for both Cartesianism and occasionalism. I call it Leibniz’s problem: How can passive matter ‘obey’ laws? How are laws executed if matter does not have what it takes to execute them? In his reply to Bayle’s criticism in 1698, Leibniz stressed that even though general laws are decrees of God, they are in need ‘of a natural means of carrying [them] out’; hence, ‘all that happens must also be explained through the nature which God gives to things.’¹¹⁶ In his ‘On the Nature itself, or on the Inherent Force and Actions of Created Things’ (1698), Leibniz gave the following argument for the need to make bodies powerful and causally efficacious. Laws are certainly God’s decrees; but for bodies to be able to execute the laws in the future, something must have been ‘impressed upon creatures’ by God which makes them capable to act according to the law. For otherwise there are no necessary connections between causes and effects; the command (God’s law) is either not binding at future moments or ‘it must always be renewed in the future.’ The command (the law), which was set in the beginning of time, is binding now and in the future only if ‘the law set up by God does in fact leave some vestige of him expressed in Things.’ And this implies that ‘there is a certain efficacy residing in things’; (and creatures in general). It is this efficacy which makes them ‘capable of fulfilling the will of him who commanded them.’ This efficacy is part of their nature and it’s a force ‘from which the series of phenomena follows according to the prescription of the first command.’¹¹⁷ Leibniz’s argument implies that both laws and inherent powers are required for the explanation of natural phenomena. But powers are individuated independently of laws; they are presupposed for the existence of necessary connections in nature in the sense that powers are required for things to obey laws and for the laws to be binding. In his letter to Hartsoeker (Hanover, 10 February 1711), Leibniz makes clear that it is not enough for the identification of a power to state a law that it obeys (or simply that there is law); what is also required is the specification of the mechanism by means of which this power acts. The mechanism is, clearly, on top of the law and given independently of it. Without the mechanism the power is ‘an unreasonable occult quality.’ He says: ¹¹⁴ Leibniz (1989b, 338). ¹¹⁷ Cf. Leibniz (1989b, 501).

¹¹⁵ Leibniz (1989b, 338).

¹¹⁶ Leibniz (1989b, 494).

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Thus the ancients and the moderns, who own that gravity is an occult quality, are in the right, if they mean by it that there is a certain mechanism unknown to them, whereby all bodies tend towards the center of the earth. But if they mean that the thing is performed without any mechanism by a simple primitive quality, or by a law of God, who produces that effect without using any intelligible means, it is an unreasonable occult quality, and so very occult, that it is impossible it should ever be clear, though an angel, or God himself, should undertake to explain it.¹¹⁸

The bottom line is that qua divine commands, laws are binding only if bodies are capable of following them necessarily—and this is grounded in their ‘impressed nature.’ Fundamentally, their impressed nature (‘the substance of things itself,’ as Leibniz put it), ‘consists in the force of acting and being acted upon.’¹¹⁹ This is ‘a primitive motive force,’ which is ‘superadded’ to extension and mass and grounds motion and action. For reasons we cannot go into here, Leibniz identifies this primitive force with a soul or substantial form.

7. Hume and Newton on Laws and Necessity Recently, there has been considerable controversy about what Hume thought about necessity. The trend is to interpret him as a sceptical realist, while the traditional view was that he was a denialist. I think that the traditional view is essentially correct: Hume denied that there is any power in nature; hence he denied that there is any necessity in nature.¹²⁰ Hume did not deny that there are laws of nature. However, these laws do not govern; nor is it the case that worldly things obey them, in any interesting sense. Laws are just the regularities themselves and nothing more. Reflecting on the question of necessity that the laws of motion allegedly have, Hume said: The degree and direction of every motion is, by the laws of nature, prescribed with such exactness, that a living creature may as soon arise from the shock of two bodies, as motion, in any other degree or direction than what is actually produced by it. Would we, therefore, form a just and precise idea of necessity, we must consider whence that idea arises, when we apply it to the operation of bodies.¹²¹

The idea of necessity, which of course Hume never doubted that we possess and that is part of the common understanding of causation, is a projection of the human mind on nature, which is conditioned by the existence of uniformity and regularity in nature: Our idea, therefore, of necessity and causation arises entirely from the uniformity, observable in the operations of nature; where similar objects are constantly conjoined together, and the mind is determined by custom to infer the one from the appearance of the other. These two circumstances form the whole of that necessity, which we ascribe to matter. Beyond the ¹¹⁸ In Newton (2004, 112). ¹¹⁹ Leibniz (1989b, 502). ¹²⁰ See Psillos 2002, chapter 1 for a discussion. ¹²¹ Hume (1748/2007, Section VIII, Part I, 82).

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constant conjunction of similar objects, and the consequent inference from one to the other, we have no notion of any necessity, or connexion.¹²²

Newton’s attitude, which of course preceded Hume’s, was quite different.¹²³ On my reading of Newton, there is power in nature but powers and laws are mutually determined—to introduce a power is to introduce a law. Interestingly, Newton did allow that there is necessity in nature; but this necessity is, from an empiricalscientific point of view, ineffable. To substantiate the point that for Newton powers and laws are mutually determined, we should briefly compare him with Descartes. In Definition III of the Principia, Newton introduces the power (vis) of matter of resisting acceleration (change of state of motion) by means of a law. He says: ‘Inherent force of matter (vis insita) is the power of resisting by which every body, so far as it is able (quantum in se est), perseveres in its state either of resting or of moving uniformly straight forward.’¹²⁴ Let’s compare this with Newton’s first law of motion: ‘Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.’¹²⁵ The Definition defines a power a body has intrinsically, viz., the power to persevere in its state of motion (i.e., the power to resist changes in its state of motion), by stating the law it obeys quantum in se est.¹²⁶ In the statement of the first law, on the other hand, Newton describes the law a body obeys by stating what happens to a body if the intrinsic power to persevere in its state of motion is the only one that is present and acting, i.e., if the body is quantum in se est. The definition has the direction from the power to the law; the law has the direction from the law to the power. A way to combine the two would be the following: Every body obeys the first law of motion (it perseveres in its state of being at rest or of moving uniformly straight forward) if and only if its vis insita is the only power acting on the body. In this sense, a power is defined by the law the body that has it obeys; and conversely, a law states what a body does in virtue of a power it possesses.¹²⁷ Descartes, by contrast, simply stated the law: ‘that each thing, as far as is in its power (quantum in se est), always remains in the same state; and that consequently, when it is once moved, it always continues to move.’¹²⁸ In his case, as we have already noted, there is no power to be defined; the law simply replaces the power. Intrinsically, i.e., quantum in se est, the body obeys the law (it always remains in the same state of motion). ¹²² Hume (1748/2007, Section VIII, Part I, 82). ¹²³ My understanding on Newton has been influenced by long discussions with Robert DiSalle. ¹²⁴ Newton (2004, 60). ¹²⁵ Newton (2004, 45). ¹²⁶ For more on the importance of Quantum in Se Est see I. Bernard Cohen (1964). ¹²⁷ As George Smith nicely put it: ‘The law characterizing a force from a physical point of view gives its “physical proportions” and assigns it to a “physical species.” Two forces are of the same physical species only if they are characterised by the same law’ (2002, 151). ¹²⁸ Descartes (1984b, 59).

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In the preface to the second edition of the Principia in 1713, which was written by Roger Cotes under the close supervision of Newton, Cotes carefully distinguished Newton’s views from those of the Aristotelians as well as from those of the Cartesians. The chief point made against the Aristotelians was that they posited sui generis powers and this was redundant and non-explanatory. Cotes wrote: There have been those who have endowed the individual species of things with specific occult qualities, on which—they have then alleged—the operations of individual bodies depend in some unknown way. The whole of Scholastic doctrine derived from Aristotle and the Peripatetics is based on this. Although they affirm that individual effects arise from the specific natures of bodies, they do not tell us the causes of those natures, and therefore they tell us nothing.¹²⁹

Note the complaint about the ‘unknown way’ on which the operations of bodies possessing a certain power depend. Newton’s emphasis on laws in defining powers was meant, among other things, to capture, and hence to explain, how powers act: they act via laws; better: to introduce a power is to introduce the law that things that possess it obey. There is no explanatory gap here. No unknown modus operandi. But Cotes goes on to criticize the Cartesian way too. As is well known, his chief point against the Cartesians in the preface was that they recourse to unfounded hypotheses and speculations about the mechanical causes of the phenomena. It transpires, however, that Cotes’s real complaint was that they do all this in order to avoid the Aristotelian pitfall of powers. They try to rectify the sui generis and nonexplanatory nature of Aristotelian powers by avoiding powers altogether and by appealing, instead, to mechanical hypotheses about matter in motion. Here is how Cotes puts the point when it comes to what Cartesians say about gravity: For either they will say that gravity is not a property of all bodies—which cannot be maintained—or they will assert that gravity is preternatural on the grounds that it does not arise from other affections of bodies and thus not from mechanical causes. Certainly there are primary affections of bodies, and since they are primary, they do not depend on others.¹³⁰

Hence, the price that the Cartesian way out comes with is either denying that gravity is a universal property of bodies or asserting that gravity is a mysterious property (power) of things since it is not explained and grounded mechanically (i.e., by means of a law-obeying configuration of matter in motion). When then Cotes states that there are ‘primary affections of bodies,’ and that being primary, these affections (gravity being one of them) ‘do not depend on others,’ he carves out precisely the middle road that Newton suggested, viz., to introduce a primary affection of matter which is neither occult (as the Aristotelians would have it) nor mysterious (as the Cartesians would have it) one would have to introduce it by means of the law it obeys,

¹²⁹ Newton (2004, 43).

¹³⁰ Newton (2004, 51).

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even if this law was not mechanical. I take it then that Cotes’s and Newton’s via media was to keep both powers (against the Cartesians) and laws (against the Aristotelians) but to claim that they are introduced hand-in-hand—especially when it comes to the primary (and hence not-further-reducible) powers of matter. Cotes makes this point (fairly) clearly when he talks about gravity being such a primary affection: ‘Among the primary qualities of all bodies universally, either gravity will have a place, or extension, mobility, and impenetrability will not. And the nature of things either will be correctly explained by the gravity of bodies or will not be correctly explained by the extension, mobility, and impenetrability of bodies.’¹³¹ Gravity is not an occult power (even though we may not know its cause) because, as Newton put it in the General Scholium, ‘it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.’¹³² Hence, the Newtonian method, as explained by Cotes, is a via media: ‘From certain selected phenomena they deduce by analysis the forces of nature and the simpler laws of those forces, from which they then give the constitution of the rest of the phenomena by synthesis.’¹³³ Newton, as is well known, did allow that there might be an unknown cause of gravity. So he might be taken to have allowed that there might be ways to identify powers independently of the laws they obey. But he was adamant that this kind of independent identification, if possible at all, should not be taken as a requirement for a legitimate appeal to powers; specifying the law that they obey is enough for scientific purposes. In an unsent letter written circa May 1712 to the editor of the Memoirs of Literature, Newton referred explicitly to Leibniz’s letter to Hartsoeker, and stressed that it is not necessary for the introduction of a power—such as gravity—to specify anything other than the law it obeys; no extra requirements should be imposed, and in particular no requirement for a mechanical grounding. He said: And therefore if any man should say that bodies attract one another by a power whose cause is unknown to us, or by a power seated in the frame of nature by the will of God, or by a power seated in a substance in which bodies move and float without resistance and which has therefore no vis inertiae but acts by other laws than those that are mechanical: I know not why he should be said to introduce miracles and occult qualities and fictions into the world.¹³⁴

There is hardly any need to relate here Newton’s three laws of motion.¹³⁵ What’s important to note is that for Newton these laws were not metaphysically necessary. As Cotes put it: ¹³¹ Newton (2004, 50). ¹³² Newton (2004, 92). ¹³³ Newton (2004, 44, emphasis added). ¹³⁴ Newton (2004, 116). ¹³⁵ For some insightful points concerning mathematical and physical characterizations of forces by Newton, see Janiak (2007). The relevant literature is, of course, vast.

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It is the province of true philosophy to derive the natures of things from causes that truly exist, and to seek those laws by which the supreme artificer willed to establish this most beautiful order of the world, not those laws by which he could have, had it so pleased him.¹³⁶

There is little doubt that laws required a law maker and this was God. But, for Newton, significantly, and unlike Descartes, God could have established different laws. Hence, the laws are metaphysically contingent; they could be different from what they are. The task then of natural philosophy is to discover the actual laws of nature—the ones God did establish. The key point here is that since God could have established other laws, finding the actual laws cannot be a matter of a priori theorizing (as Descartes suggested) but of a broadly empirical investigation. Indeed, Cotes says: From this source [God’s perfectly free will], then, have all the laws that are called laws of nature come, in which many traces of the highest wisdom and counsel certainly appear, but no traces of necessity. Accordingly we should not seek these laws by using untrustworthy conjectures, but learn them by observing and experimenting.¹³⁷

How can it be that there are no traces of necessity in the laws of nature? If laws were metaphysically necessary they would be known a priori; hence independently of experience. But for Newton and Cotes there cannot be a priori knowledge of laws of nature, since the laws are free choices of God. And if laws where metaphysically necessary they would not be the free choice of God. Hence there is a dilemma: either the claim is that laws hold with metaphysical necessity, but then this would not make the laws the free choice of the author of the universe; or the claim is that God was free in the choice of laws, but then they cannot be known a priori. The first horn is taken by Descartes; the second by Newton. Cotes puts the point thus: He who is confident that he can truly find the principles of physics, and the laws of things, by relying only on the force of his mind and the internal light of his reason should maintain either that the world has existed from necessity and follows the said laws from the same necessity, or that although the order of nature was constituted by the will of God, nevertheless a creature as small and insignificant as he is has a clear understanding of the way things should be.

Part of the point is that knowing the laws of nature a priori would be possible only on two conditions: either if there could be only one way the laws could be; and hence the laws were metaphysically necessary; or, should the laws be metaphysically contingent, if the human mind had the capacity to latch onto them independently of experience. Both conditions are denied by Newton. So to the question: ‘What makes the laws laws, ultimately?’ Newton’s answer is: the will of God. Given that the laws are grounded directly in the will of God, they cannot but be in some sense necessary: once God wills them, they cannot but hold and govern the behaviour of things. But this is not metaphysical necessity but conditional or natural necessity. ¹³⁶ Newton (2004, 52).

¹³⁷ Newton (2004, 57).

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In this sense, for Newton there are naturally necessary connections in nature, expressed by the fundamental laws of nature, but the only way to find them out is empirically (and hence fallibly). Newton, then, unlike Hume, takes it to be the case that laws are not mere regularities; however, he takes it that they can be known only as (mathematically characterized) regularities. This leads to a drastic reconceptualization of laws of nature as primarily mathematical principles which characterize the basic structure of the world and place constraints on the explanation and description of natural processes, but are known empirically and not a priori. In his unsent letter to Cotes, in March 1713, Newton noted: I like your design of adding something more particularly concerning the manner of philosophizing made use of in the Principia and wherein it differs from the method of others, viz. by deducing things mathematically from principles derived from phenomena by induction. These principles are the three laws of motion. And these laws in being deduced from phenomena by induction and backed with reason and the three general rules of philosophizing are distinguished from hypotheses and considered as axioms. Upon these are founded all the propositions in the first and second book. And these propositions are in the third book applied to the motions of the heavenly bodies.¹³⁸

8. Concluding Thoughts During the seventeenth century, there is a clear shift from powers as regularity enforcers to laws as regularity enforcers. Medieval powers yielded a bottom-up necessity: laws were metaphors for naturally necessary connections among powerful substances. But during the seventeenth century, laws replaced natural powers as principles of connection. Laws yield a top-down necessity. They derive their necessity from a law giver and they determine how things must behave in the world; that is, they ground and explain the regular patterns there are in the world. Laws, in this sense, are ‘behind’ the regularity there is in the world and ground it. As we have seen, however, laws kept a key feature of powers: they imposed necessary connections in nature, though there was a shift in how exactly necessity was conceived. Cartesian laws hold with metaphysical necessity; not so for the rest of the thinkers we examined: laws hold with natural necessity and impose patterns of naturally necessary connections in nature. The shift from powers to laws came with a tension, which was accentuated by the fact that matter was taken to be inert and causally inactive. This tension is captured by the following two questions: How can things obey laws if they do not have the power to do so? How can a powerless matter execute the laws? Laws are supposed to govern the behaviour of things; things in some sense obey them and hence execute them. But as Leibniz forcefully argued, for matter to be ¹³⁸ Newton (2004, 109–10).

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capable of executing a law, matter must have suitable powers. With Leibniz powers re-enter the stage as that in virtue of which bodies are subject to laws and able to execute/obey the laws. Laws then seem to require powers to enforce necessary connections in the world. But how do objects acquire their powers? For Leibniz, as we have seen, powers are identified independently of laws. But two other answers to this question became available. One was Hume’s, and the other was Newton’s. For Hume, there are no powers. The laws of nature are regularities. For Newton, powers and laws enter the world hand in hand, as it were. Unlike Hume, Newton was far from accepting that there is no natural necessity. Still, of this necessity nothing can be known except whatever is given to us as (mathematically characterized) regularities. If there is natural necessity in nature (Newton thinks there is), it can never be found out. But let’s compare Newton with Hume once more: If God is left out of the picture, and if natural necessity cannot be otherwise grounded in (the independently posited) natures or powers, all we are left with are regularities which can be known only as (mathematically characterized) regularities (and never qua natural necessities). These, plausibly, are the laws of nature.¹³⁹

¹³⁹ I would like to thank Lydia Patton and Walter Ott for their immense patience and encouragement. Walter’s work has been a great source of inspiration and ideas. Many of the ideas expressed in this chapter were extensively discussed with Robert DiSalle and Stavros Ioannidis, whom I thank wholeheartedly. Earlier versions of the chapter were presented in an invited talk at the Rotman Institute of PhilosophyEngaging Science in June 1016; as part of a symposium (in absentia) in the HoPoS Conference in Minneapolis in June 2016; and in the inaugural Conference of the POND-Philosophy of Science around the Mediterranean in Jerusalem in September 2016. My thanks go to several members of the audiences for comments and questions.

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6 Laws and Ideal Unity Angela Breitenbach

1. Introduction The lawful unity of nature plays a central role in Kant’s philosophy. According to Kant, all natural phenomena are without exception unified by a set of a priori laws. Kant develops this claim first and foremost in the Critique of Pure Reason. There, he sets out the fundamental laws of nature in general. They include, for example, the principles that substance persists throughout all change and that every change of the states of a substance has a cause.¹ On Kant’s account, these laws do not merely describe the regularities that obtain in the world, but they are necessary principles. The necessity they express is not given independently in the objects themselves, but it is grounded in the nature of the human intellect. As Kant puts it, human understanding ‘prescribes’ the laws to nature, and only things that stand under these laws are proper objects of human cognition.² We can thus know a priori that all of nature in general is governed, for example, by the principles of substance and causality Kant spells out in the first two ‘Analogies of Experience.’³ The universal laws of nature in general do not determine fully the particular character of specific phenomena. Kant thus argues that different kinds of substances, and different kinds of causal relations between the states of those substances, are furthermore governed by a set of more specific laws. Kant is most explicit about this second type of law in the Appendix to the Transcendental Dialectic of the first Critique and in the introductions to the Critique of Judgment. He suggests, for example, that different causal relations are determined by their own ‘rules,’ the ‘particular (empirical) laws of nature.’⁴ These particular laws do not have full generality but determine the different kinds of things that are the objects of the special sciences. By contrast with the a priori laws of nature, the more specific laws cannot therefore be known by a priori reflection on the conditions of the objects of ¹ A182/B224 and A189/B232. All references to Kant are to the volume and page numbers of the Akademie edition (Kant 1900 ff.), except in case of the Critique of Pure Reason, which is cited by reference to the A and B pagination of the original 1781 and 1787 editions. Translations are from Kant (1992, 1999, 2000, and 2007), unless indicated otherwise. ² Prolegomena, 4:320. ³ A176/B218. ⁴ Critique of Judgment (CJ) 5:183.

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experience in general. Instead, knowledge of these laws requires experience of the particularities of specific phenomena. According to Kant, natural phenomena are thus determined by the fundamental a priori laws as well as the particular and empirical laws of nature. Kant’s arguments for the a priori laws have received ample coverage in the secondary literature. But his further claims about the empirical laws raise special questions, which have been the concern of several commentators in the more recent literature. As is well established, Kant agrees with Hume that experience cannot ground knowledge of necessity. How, then, are we to construe these empirical laws? How, in particular, can they be both necessary and known empirically? Further questions arise for the conception of unity that Kant associates with this second set of laws. In the Appendix to the Transcendental Dialectic, he contrasts the ‘collective unity’ of cognition under the empirical laws with the ‘distributive unity’ of cognition under the a priori laws.⁵ A distributive unity extends to any possible instance of a given unifying principle. The unity of cognition made possible by the a priori laws is of this kind; all empirical cognitions are united as the indefinitely many possible instances of these laws.⁶ By contrast, the empirical laws of nature generate what Kant characterizes as a collective unity. The collective unity, as he puts it, is ‘not merely a contingent aggregate, but a system interconnected in accordance with necessary laws.’⁷ It includes a plurality of diverse cognitions, determined by a plurality of diverse laws, that stand in a determinate relation to one another in a single and complete whole. The notion of a collective, or systematic, unity of cognition in accordance with empirical laws is problematic primarily because of the epistemic status Kant ascribes to it. Although both types of unity, distributive and collective, play important roles in our cognition of nature, they perform these roles in very different ways. On one side, we can know with full certainty that all empirical cognitions are unified by the a priori laws of the understanding. On the other side, we have no insight whatsoever into the collective unity of our cognitions of the phenomena under the more specific, empirical laws. The notion of a systematic unity is not ‘the concept of an object,’ as Kant puts it, but ‘a mere idea’ without determinate application to the phenomena.⁸ Kant nevertheless claims that the idea of the collective unity of nature has ‘an excellent and indispensably necessary regulative use.’⁹ It serves as a heuristic ideal that guides research into the phenomena and the laws that govern them.¹⁰ ⁵ A582/B610 and A644/B672. ⁶ It is worth noting here that the a priori laws generate a distributive unity both of cognition and of the objects of cognition. For each empirical cognition it is true that it is determined by the fundamental laws of the understanding; all empirical cognition is of (some kind of) enduring substances and (some kind of) causal relation between the states of these substances. Since, on Kant’s account, the a priori laws of the understanding are, furthermore, constitutive of cognition as well as of the objects of cognition, the same holds true for these objects. ⁷ A645/B673. ⁸ A645/B673 and A647/B675. ⁹ A644/B672. ¹⁰ As in the case of distributive unity, the collective unity pertains to cognition as well as to nature. Kant at first sets out this idea as a ‘logical principle’ (A650/B678), or ‘methodological device’ (A661/B689).

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ANGELA BREITENBACH

The further questions Kant’s idea of the systematic unity of cognition raises thus concern the relation of this idea to the empirical laws. Why must we presuppose the idea of systematic unity in the search for these laws? What, in particular, is the cognitive function of the idea for our knowledge of the laws given that we cannot achieve the systematic unity of cognition itself? My aim in this chapter is to sketch an interpretation of Kant’s conception of empirical laws that accounts for the centrality Kant ascribes to the idea of systematic unity. I specifically explore the epistemological question of how we can know empirical laws, given that they are construed as having genuine necessity. I argue that, in most cases, we cannot know such laws by derivation from the a priori laws of nature in general. Instead, we can discover empirical laws by reflection on particular phenomena, where this reflection relies on the regulative idea of the collective unity of nature. I suggest that although unifying reflection of this kind cannot ground scientific knowledge in the strict sense of the term, it can lead to more or less unified cognition of the empirical laws of nature. I argue for this reading of empirical laws in three main steps. I begin, in Section 2, by reviewing three recent interpretations, which differ significantly in their accounts of the necessity of the laws and our knowledge thereof. I show that while all three readings enjoy some textual support, neither is without problems. I develop my proposal in the subsequent two sections. In Section 3, I sketch an account of empirical reflection that relies on the idea of systematic unity as a means for discovering particular instantiations of the a priori laws of nature. In Section 4, I draw out some of its implications. I show that empirical reflection limits the possibility of scientific knowledge in the strict sense, but makes possible a kind of cognition that allows for hope in the progress towards knowledge. I argue, moreover, that the resulting conception of the laws suggests a non-reductive conception of unity, which allows for a non-homogeneous plurality of laws. I conclude that Kant’s conception of the lawful unity of nature is interestingly different from other conceptions of unity more commonly associated with Kant.

2. Three Interpretations Revisited Interpretations of Kant’s conception of empirical law that have dominated the recent literature may be divided into three broad camps, following James Messina’s helpful classification. I refocus Messina’s survey for the purpose of this chapter by

The idea instructs us how to go about extending cognitions, and advancing our understanding of the objects of cognition. Kant also points out, however, that the idea of systematic unity has at the same time a ‘transcendental’ character (A651/B682). If the methodological principle is reasonably to be employed in the search for unified cognition, we must presuppose that the objects are such that they can be cognized by this method. Otherwise, Kant claims, reason would ‘set as its goal an idea that entirely contradicts the arrangement of nature’ (A651/B682).

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considering the role that the idea of systematic unity plays in these accounts of empirical law.¹¹ First, the Best System Interpretation (BSI), proposed by Philip Kitcher among others, places the idea of systematic, or collective, unity center stage.¹² According to BSI, the particular laws of nature are those empirical generalizations that would figure in the best systematization of the empirical data at the ideal end of inquiry. The best system, or in Kantian terms the systematic unity of our cognitions of nature, is what confers the status of a necessary law on empirical regularities. There can be no genuine empirical law independent of such a system, and we can know the laws only insofar as they form part of our best systematization of the empirical data. According to BSI, systematic unity is thus constitutive of both the necessity and our knowledge of the laws. On a second and competing interpretation, defended most prominently by Michael Friedman, the systematization of empirical data provides a method for a first approximation to knowledge of particular laws.¹³ Systematic unification can guide empirical inquiry into exceptionless generalizations. However, it cannot, on its own, guarantee cognitive access to their lawfulness. For genuine knowledge of particular laws, we must derive those laws from the a priori laws of nature together with the relevant empirical content. On this Derivation Account (DA), particular generalizations are necessary laws, and can be known as such, only if they can be derived in this way. The systematic unity of nature, by contrast, is an idea that offers only a preliminary guide to empirical laws. A third interpretive approach has most recently been developed by a growing number of authors.¹⁴ The key proposal is that, according to Kant, empirical laws are necessary governing principles that obtain by virtue of the particular natures of things.¹⁵ According to this Necessitation Account (NA), empirical laws are necessary independently of our ability to derive these laws from the a priori principles of nature in general—or, indeed, to include the laws in our best systematization of the ¹¹ Messina (2017). As this refocusing will make clear, I believe that Messina’s own account and others he sides with unduly downplay the relevance of the idea of unity. Beyond the three camps mentioned here, there is another literature relevant for the questions of this chapter, focused more explicitly on the idea of systematic unity and less centrally on the nature of empirical laws. See, e.g., Guyer (1997), Geiger (2003), and Ginsborg (2017). ¹² Kitcher develops his interpretation in a series of papers (see, e.g., 1986 and 1994). Buchdahl (1969, 1992), Brittan (1978), and Allison (1996) are also often associated with this interpretation. ¹³ Friedman (1992, 2013, and 2014). ¹⁴ This third class includes a variety of interpretations such as Watkins (2005), Kreines (2009, 2017), Massimi (2014, 2017), Messina (2017), and Patton (2017). ¹⁵ Proponents of the third approach sometimes contrast their ‘bottom-up’ account with ‘top-down’ models (see, e.g., Messina 2017). According to the latter, the necessity of particular laws is generated by derivation from the a priori laws, that is, from above. The former construe necessity as grounded in the particular natures of things, that is, from below. A similar distinction was previously associated with the debate between BSI and DA. By contrast with the more recent metaphysical distinction, however, the earlier contrast was primarily focused on the epistemological question of our knowledge of the laws. For a classification of Early Modern conceptions of law into bottom-up and top down accounts, see Ott (2009).

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empirical data. In this respect, NA downplays the role of the idea of systematic unity. Some proponents of NA have argued that we may nevertheless attempt to systematize empirical phenomena with the aim of discovering particular laws. But such systematization will account neither for the necessity nor for our knowledge of the laws. Since the necessity of particular laws is grounded in the essential natures of things, which are inaccessible to cognizers like us, these interpreters have concluded that empirical laws are for us in principle unknowable.¹⁶ All three readings have textual support. Proponents of BSI can primarily point to passages in the Appendix to the Transcendental Dialectic and the introductions to the Critique of Judgment. There, Kant stresses the role of reason and reflecting judgment that, in the search for cognition of empirical laws, necessarily follow principles of systematic unity.¹⁷ DA finds specific support in texts of the Transcendental Deduction of the Categories and the Metaphysical Foundations of Natural Science. There, Kant argues that necessity and strict universality can be grounded not in empirical generalizations but only in the a priori principles of the understanding.¹⁸ NA, finally, can rely on passages in the first and third Critiques and on Kant’s preCritical texts. Particularly in the Critique of Judgment Kant argues that different natures have causal powers acting in accordance with many different rules into whose necessary status, however, we have no insight.¹⁹ All three interpretations also face a number of problems. The difficulty with BSI is that it seems unclear how inclusion in a system can confer anything but a weak form of necessity on generalizations that would otherwise be contingent. According to BSI, to be a law is to belong to an ideal systematization of empirical generalizations. But how could systematized generalizations be said to determine or govern the phenomena? Moreover, as critics have noted, BSI does not account for the explanatory role of the laws of nature.²⁰ Generalizations, even if part of an ideal system, would not explain why the observed regularities obtain. They would fail to elucidate the necessary conditions that determine the phenomena. According to DA, by contrast, particular laws have the same strong necessity as the a priori laws. They are necessary governing principles that can play the required explanatory role. They can not only provide a description of regular occurrences but elucidate the necessary conditions of particular phenomena. As critics have noted, however, the problem with DA is that few generalizations we ordinarily count as laws seem actually derivable from the a priori laws of the understanding.²¹ Friedman

¹⁶ Kreines (2009). See also Messina (2017). By contrast, Massimi (2017) has argued that we can come to know the necessity of the laws even if their grounds are inscrutable to us. In this chapter I agree with Massimi’s conclusion that cognition of the laws is possible. However, as I argue below, my reasons are rather different from her dispositional essentialist reading of Kant’s account of laws. ¹⁷ E.g. A680/B708ff. ¹⁸ E.g. A91/B124. ¹⁹ E.g. CJ 5:183. On support for NA in the pre-Critical Kant, see Watkins (2005, chapter 2), Massimi (2014) and Messina (2017). ²⁰ See Kreines (2009) and Messina (2017). ²¹ See Kitcher (1994).

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elaborates on the case of Newton’s laws of motion.²² But other generalizations, Bernoulli’s principle of fluid dynamics or Hauksbee’s law of gases, for example, present unpromising candidates for successful derivation in this way. One might reply that this may not be a problem in and of itself, since there might simply be very few empirical laws. But a more general question nevertheless arises for DA. Why should we expect the results of our systematizing activities to converge with the principles derivable from the a priori laws? In other words, why should we have ‘the well-founded hope,’ as Friedman puts it, that our attempts at systematizing the empirical data will progress towards the kind of derivation necessary for grounding genuine lawfulness?²³ Compared with DA, proponents of NA have offered a neat explanation of how Kant can hold that empirical laws are necessary, while also accounting for the epistemic restrictions that stand in the way of our deriving those laws from a priori principles. But the proposal raises further questions. If the empirical laws of nature are necessitated by the particular natures of things, yet those natures are in principle inaccessible to us, then how can we ever come to know any empirical laws as genuinely necessary? Kreines’s and Messina’s answer is that we cannot. On their account, there are—or, perhaps, may be—particular necessities that we cannot ever come to know.²⁴ However, given Kant’s extensive discussion of empirical laws in the context of his account of cognition, it would be somewhat surprising if our principled ignorance of particular laws were his last word on the matter. More specifically, NA would be open to a similar criticism as DA. It would leave unclear how any scientific inquiry, in particular attempts at systematizing empirical phenomena, could reasonably be regarded as aiming at, or ‘approaching,’ knowledge of empirical laws.²⁵ In their accounts of particular laws, the three interpretations have thus taken different routes through the difficulty of explaining how laws can be necessary and knowable by experience. Proponents of DA and NA have given good arguments for the necessary status of particular laws. They have shown that empirical laws of nature are not simply generalizations that describe what actually happens, but necessary principles that explain why something has to happen. But they have thereby downgraded the centrality of the idea of systematic unity and, with it, the possibility of our knowledge of the laws. For DA, knowledge is effectively restricted to only a few laws; ²² See Friedman (2013). The laws of matter, spelt out in Metaphysical Foundations of Natural Science (MFNS), are a special case. Since nature is as a matter of fact material, they are valid for every possible natural phenomenon. This is why, on Kant’s account, they can be derived from the a priori laws together with the empirical concept of matter. It is unclear, however, why we should hope that other, more specific laws are derivable in the same way. On the laws of the MFNS, see also Stang (2016, chapter 8). ²³ Friedman (2014: 553). ²⁴ According to Kreines (2009: 542) it is an open question whether Kant asserts knowledge that there are laws governing specifically distinct natures, even if he denies knowledge of what those laws might be. ²⁵ Kreines (2009: 537). Having denied knowledge of empirical laws, Kreines wants to preserve ‘progress without knowledge of natural laws’ (534).

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and, according to some proponents of NA, knowledge of most empirical laws is impossible in principle. BSI, by contrast, has upheld the possibility of our knowledge of empirical laws, achieved through the systematization of empirical generalizations. But it has done so only at the cost of downgrading the necessity of these laws. The laws we can know, according to BSI, are necessary in a weaker sense than the a priori principles. The survey of these three interpretations lays bare a tension between attempts to account for the necessity and knowability of the empirical laws of nature. In the following I explore whether we can do more to alleviate this tension on Kant’s account. I suggest that we can take seriously the claim, shared by DA and NA, that the necessity of particular laws is grounded in conditions that lie beyond any possible experience, while offering a more satisfactory account of our cognition of these laws.

3. Empirical Reflection and the Search for Laws Proponents of DA and NA have rightly pointed out that the necessity of empirical laws cannot be grounded in empirical regularities but only in conditions that, to us, are either accessible a priori or not accessible at all. As Kant puts it in the Transcendental Deduction, ‘appearances may well offer cases from which a rule is possible in accordance with which something usually happens, but never a rule in accordance with which the succession is necessary.’ Such a necessary rule ‘must either be grounded in the understanding completely a priori or else be entirely surrendered as a mere fantasy of the brain.’²⁶ Kant also often speaks as if we were acquainted with the particular laws of nature. For example, he describes the delight we feel when we encounter ‘systematic unity among merely empirical laws.’²⁷ He seems to take it as a given that we have some cognitive access to these laws. But how is such access possible on the assumption that empirical laws have genuine necessity? According to Kant’s account of the fundamental laws of nature, we know a priori that all phenomena are governed by necessary principles. For example, we know a priori that every change of the states of a substance has a cause and stands under some necessary causal law. By contrast, we do not know a priori which cause brings about which effect or which empirical regularity is a genuine law. Such further insight cannot be had a priori. I argue that it can be achieved empirically insofar as empirical inquiry fulfils two conditions. First, empirical inquiry is guided by the a priori laws and thus constitutes the investigation into particular instantiations of these laws. Second, empirical inquiry is guided by the idea of systematic unity and, if this idea could be fully realized, would result in knowledge of the particular laws. I suggest that through this type of empirical reflection, guided both by the a priori laws and the idea of systematic unity, we can achieve cognition of empirical laws. The right model for ²⁶ A91/B123f.

²⁷ CJ 5:183.

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our cognitive access to these laws is not derivation from a priori principles but reflection on the phenomena. In fleshing out this reflection model as an account of our cognition of particular laws, we thus need to pay attention to both of these aspects, guidance by the a priori laws and presupposition of the regulative idea of unity. First, consider the guiding function of the a priori principles for empirical reflection. I focus here on the principles of relation, Kant’s so-called Analogies of Experience.²⁸ They determine that any appearance stands to some other appearance in relations of substance-accident, cause-effect, and the interaction of agent and patient. On Kant’s account, these principles are constitutive of empirical cognition and of the objects of cognition. However, they do not automatically churn out particular substance-accident, causeeffect, or agent-patient relations when fed empirical input. They do not determine which appearances stand to each other in these three types of relation. By contrast, the Analogies are constitutive of empirical cognition indirectly. They guide us in reflecting on the phenomena with the aim of identifying the relevant relata in experience, and of discovering the empirical laws that govern these relata.²⁹ The Second Analogy, for example, does not specify the cause of the increasing temperature of a stone in the sun. But it states that there must be some appearance that caused the temperature to change. It thereby tells us that there is some particular causal law in accordance with which the change occurred. Moreover, the Second Analogy guides the search for particular laws by directing us to reflect on appearances in accordance with the a priori concepts of cause and effect. It instructs us to search for an appearance A that stands to a given appearance B just as a cause stands to its effect. It guides us, as Kant puts it, in ‘combining appearances . . . according to an analogy with the logical and universal unity of concepts’—in this case, the a priori concepts of cause and effect.³⁰ The Analogies of Experience thus have both a constitutive and a regulative function.³¹ They constitute empirical cognition by determining the form of the particular

²⁸ The story will be different for how the mathematical principles are applied in experience. Since these principles do not on their own imply an account of necessity, however, I shall set these principles aside here. More will have to be said on another occasion about the principle of modality, which play a special role in the constitution of empirical cognition for Kant. ²⁹ As Guyer (1987, 69) puts it, the analogies of experience do not ‘uniquely determine’ a particular object, but only ‘send one looking in the right direction.’ ³⁰ A181/B224. In line with this reading, the Analogies of Experience carry their name for good reason. They tell us to construe the relationship between two appearances in analogy with the relationship between the a priori concepts. ³¹ See A180/B222f. Also A236/B296 and A664/B692. Buchdahl (1969, 651) has introduced the distinction between ‘transcendental and empirical levels’ of causation. By contrast with this two-levels reading, I believe that the regulative and constitutive dimension of the analogies of experience are more closely interdependent than Buchdahl recognizes. This is because the recognition in experience of an order that can legitimately be subsumed under the relational principles requires reflection guided by these principles. The constitutive character of the principles thus presupposes their regulative employment. In this, I agree with the related claims made by Longuenesse (1998) and Ginsborg (2006a, 2006b).

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laws that govern the phenomena as consisting in substance-accident, cause-effect, or agent-patient relations. But they can perform this constitutive function only by fulfilling a second, regulative role: they guide empirical inquiry by indicating the types of universal sought in experience as having the forms of substance-accident, cause-effect, or agent-patient relations.³² Reflecting on the phenomena in accordance with the Second Analogy, for example, is thus a means for investigating specific causal relations and the laws that govern them. In this way, the reflective search for empirical laws is a search for the particular instantiations of the a priori principles. And yet the regulative function performed by the a priori principles is only part of the picture. The reflective search for particular laws also requires guidance by the regulative idea of systematic unity. On Kant’s account, the only way to ascertain whether we have correctly identified a necessary law that governs the change of the states of a substance is to compare the given particular with other similar cases. By comparing and contrasting particulars, we can examine whether the same causes have the same effects, whether different effects have different causes, whether we have thus identified a candidate empirical law, and whether this law fits together with other candidate empirical laws. However, Kant’s account also entails that the empirical data is never complete. Further evidence may always suggest that we have picked out the wrong generalization and have thus identified the wrong cause of some given effect. The empirical inquiry, described so far, might prompt all sorts of skeptical worries. Kant thus argues that, in order to regard empirical data as evidence for or against a particular law, we must presuppose it as a regulative idea that the evidence we have is part of a complete whole of cognitions. In other words, we must follow the regulative principle that the empirical data we have gathered contribute to completing the totality of what there is to be found out about the phenomena. Only if we assume the unity of cognitions in this sense, can we take particular empirical inquiry as getting us closer to our knowledge of the necessary laws that govern the phenomena. Or, to put the point in the terms of the third Critique, regarding empirical data as evidence for a particular law, we must presuppose the regulative principle of the ‘purposiveness of nature’ for our understanding.³³ We have to go about our empirical inquiries as if nature were conducive to cognition by the epistemic means we have at our disposal. Only if we take nature to be purposive for our understanding in this sense, can we regard reflection on the phenomena as resulting in cognition of the particular laws of nature.³⁴

³² In unpublished work, Matthew Boyle has argued that the categories characterize the form that acts of reflection on the sensible manifold can take. In a similar manner, I suggest, the relational principles can be understood as guiding reflection by specifying the universal concept or law at which such reflection aims. ³³ CJ 5:181. ³⁴ The idea of unity should therefore not be mistaken for the belief in the uniformity of nature that Hume famously takes to underlie empirical inquiry. Moreover, the search for particular laws guided by the idea of unity is continuous with the intellectual activity of conceptualizing the relevant data. In both points, I concur with Ginsborg (2017).

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Kant thus suggests that ‘the systematic unity of the understanding’s cognitions . . . is the touchstone of truth for its rules.’³⁵ By analogy with a tool for assaying the purity of precious metals, Kant claims that the systematic unity of empirical cognitions functions as a test for the truth of the regularities we have identified as empirical laws. If all cognitions were systematically unified, we would have achieved complete understanding of the phenomena. Under these ideal conditions, we would have cognition of the complete set of conditions of any given occurrence. We would have all the possible evidence relevant for identifying whether a particular generalization is indeed the instantiation of an a priori law. It is this regulative idea of systematic unity, Kant argues, that must be presupposed if we are to take a particular empirical relation as instantiating a necessary law. The empirical inquiry that is guided by the idea of unity is thus not simply an act of systematizing empirical generalizations according to certain logical principles. It is not, as BSI has it, an attempt to arrange rules, which we had previously formulated on the basis of empirical data, in the best overall system. On the Kantian conception, empirical inquiry should rather be construed as a reflective act, regulated both by the idea of systematic unity and by the a priori principles. It is a process in which the idea of unity guides us in comparing and contrasting particulars in the search for general laws, and in which the a priori principles determine the necessary form of the concepts and laws we seek. I suggest that it is because empirical inquiry is informed, in this way, by the a priori principles as well as the idea of systematic unity, that we can regard our systematizing activities as the search for particular necessary laws.

4. Ideal Unity and Our Knowledge of Laws I have argued that we can regard empirical reflection as the search for particular laws insofar as such reflection is guided both by the a priori principles and by the idea of systematic unity. But the foregoing discussion raises an important question about the possibility of our knowledge of the laws. If our search for knowledge of particular laws presupposes the idea of unity, yet the full realization of this idea can never actually be achieved, is not the knowledge of empirical laws, too, unachievable? Should we conclude, as some proponents of NA have done, that knowledge of particular laws is in principle impossible? As Kant argues, knowledge requires not only sufficiently articulated empirical cognition, but also assent to a judgment and justification of its truth.³⁶ As he puts it, when ‘taking something to be true is both subjectively and objectively sufficient it is called knowing.’³⁷ Full justification, however, would require the unity of cognition which, as Kant is keen to stress, cannot in ³⁵ A647/B675. ³⁶ A822/B850. On Kant’s conception of knowledge, and its relation to cognition, see Chignell (2014a) and Watkins and Willaschek (2017). ³⁷ A822/B850.

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principle be achieved. Kant’s conception of our knowledge of the laws thus appears exceedingly demanding and, ultimately, impossible to attain.³⁸ To this extent, then, I agree with the conclusion reached by some proponents of NA.³⁹ But this is not all Kant has to say on the matter. Despite his important limitation to our knowledge of empirical laws, Kant is keen to stress the contrast between knowledge in the strict sense and ‘common cognition.’ As he points out, in many cases where we take ourselves to know, what we have is an ‘aggregate of cognitions’ and, perhaps, sufficiently unified cognition relative to a given purpose.⁴⁰ In the introduction to the Metaphysical Foundations, Kant furthermore contrasts knowledge required for ‘proper science [which] is only that whose certainty is apodictic’ with ‘improper’ knowledge, the kind of cognition ‘that can merely contain empirical certainty.’⁴¹ In addition to the strict notion of scientific knowledge, Kant thus makes room for knowledge in a more inclusive sense. This ‘improper’ knowledge, or common cognition, requires conscious representation of what is given in sensibility by means of concepts ‘that pertain to objects.’⁴² Common cognition may be more or less unified. It may have more or less content, and may include more or less information about the conditions of the phenomena. It can be improved in this regard by the unification of cognitions.⁴³ Moreover, common cognition does not require any further justification of its truth. It is thus never entirely beyond doubt, and we must always be ready to accept that even cognition that is sufficiently detailed for our purposes will have to be revised. Nonetheless, common cognition is all we often have when we take ourselves to know, and it may reasonably be regarded as a form of knowledge in the loose sense of the term.⁴⁴ With his account of common cognition in place, Kant can thus regard empirical laws as lying within our cognitive reach and as making empirical explanations possible. He can claim that we can cognize the empirical generalizations we have discovered through our systematizing inquiry ‘as laws (i.e., as necessary),’ even though we cannot know them in the strict sense.⁴⁵ Moreover, the relation between common cognition and strict knowledge accounts for the possibility of ‘progress’ or ‘approximation to knowledge’ of empirical laws.⁴⁶ The search for more and

³⁸ This is why in the MFNS, Kant focuses on laws that can be constructed a priori given the empirical concept of matter. Our knowledge of these laws does not, therefore, face the same problems as the more specific empirical laws. And this is also why he makes his famous claim that ‘in any special doctrine of nature there can be only as much proper science as there is mathematics therein’ (4:470). ³⁹ By contrast, Chignell (2014a, 592) presents a more inclusive conception of the knowledge of laws, claiming that ‘our shared background knowledge of nature and its laws’ plays the crucial role in grounding knowledge claims. ⁴⁰ See Kant’s account of the perfection of cognition in the Jäsche Logic (JL) 9:72. ⁴¹ MFNS 4:468, my translation. ⁴² A78f./B104f. ⁴³ See JL 9:64. ⁴⁴ In Breitenbach (2011) I have shown how Kant characterizes this lower form of knowledge as ‘applied rational cognition.’ ⁴⁵ CJ 5:184. ⁴⁶ Kreines (2009, 534, 537).

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more unified cognition is guided by the very idea whose realization would ground scientific knowledge. Improving, or unifying cognition through empirical inquiry, thus asymptotically approaches knowledge of the laws.⁴⁷ The systematizing search for cognition both advances our ‘improper’ knowledge of the empirical laws and would—if it could be completed—be a positive test of their truth. It is because of this intrinsic connection of achievable cognition, or low-grade knowledge, and unattainable knowledge in the strict sense, that we can speak of epistemic progress. Moreover, because of this intrinsic connection we also have rational grounds for ‘hope’ that the generalizations resulting from our systematizing inquiry will be the laws known under ideal conditions.⁴⁸ Ordinary knowledge, epistemic progress, and rational hope all have a place in Kant’s account of empirical laws. The reading I have proposed shows that the strict limitations on our knowledge of the laws, put forward by proponents of NA, needs some qualification. Although we can never be absolutely certain that our empirical regularities are necessary, we can have more or less unified cognition of particular laws, or knowledge in the loose sense of the term. The idea of unity thus performs a key role in Kant’s account of our knowledge of the laws and proponents of BSI are right in ascribing the idea of unity prominence. But this does not entail that the systematic unity of our cognitions also confers necessity upon empirical generalizations. As I have suggested, the idea of systematic unity is not constitutive of the necessity of particular laws but only regulative for our cognition thereof. The interpretation I have proposed has further implications for the idea of unity presupposed in our search of empirical laws. The special character of Kant’s position, on my reading, can be brought out by contrasting it with DA. The conception of the collective unity associated with DA is that of a whole in which the particular laws can be derived from the most general laws together with particular empirical data. It is natural, I think, to link this conception with a reductionist notion of unity. On this conception, our cognition of nature is unified by a hierarchy of laws. More specific laws determine our cognition of more complex phenomena on the lower levels of the hierarchy, and stand under more general laws, which determine our cognition of phenomena on higher levels. The unity of these laws is ultimately fixed at the top by the highest, a priori, laws of nature. More specific laws are thus ultimately derivable from the most general laws. The reading I have proposed in this chapter shows, I believe, that Kant’s conception of unity does not entail a reductionist conception of unity.⁴⁹ On my reading, the particular laws must be thought of as forming part of a collective, or systematic, unity. That is, the laws cannot be regarded as a plurality of entirely independent

⁴⁷ See A663/B691. ⁴⁸ Friedman (2014, 553). On Kant’s conception of rational hope see Chignell (2014b). ⁴⁹ For further considerations of a non-reductionist account of unity inspired by Kant’s account, see Breitenbach and Choi (2017).

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necessities,⁵⁰ but they must be construed as standing in determinate relations to each other that are in principle understandable to us. This unity of cognition under the empirical laws is not fixed by the a priori laws at the top. It is rather informed by these laws from within.⁵¹ It therefore does not imply a single hierarchical form but may, instead, consist of a plurality of hierarchies, ordered according to a plurality of general governing principles. As Kant’s claim that the study of biology is necessarily guided by teleological principles suggests, for example, different areas of inquiry may be carved out by irreducibly different regulative principles.⁵² The idea of the collective unity of nature implied by my reading may thus make room for a plurality of irreducible laws. Although Kant suggests that ‘sameness of kind is necessarily presupposed in the manifold of a possible experience,’ he immediately adds that ‘we cannot determine its degree a priori.’⁵³ We must search for more and more general laws of nature, on Kant’s account, but we cannot know in advance how far our unifying activities will reach. Moreover, Kant stresses that we need look not only for more and more general laws but also for the more specific principles that determine the differences between the phenomena. In addition to the principle of homogeneity, the principles of specification and affinity are equally valid. They direct us to search not only for more general principles but also for more specific laws and for the connections between them.⁵⁴ Only by investigating the laws that govern the common as well as the diverse features of the phenomena can we fill in the system of cognitions, and thus approximate to knowledge of the whole. Ultimately, Kant thinks that this unity cannot be construed hierarchically, but must be understood as having teleological and organic structure. As Kant puts it in the Critique of Judgment, ‘by means of the example that nature gives in its organic products, one is justified, indeed called upon to expect nothing in nature and its laws but what is purposive in the whole.’⁵⁵ While an analysis of this further thought lies beyond the scope of this chapter, this second implication of my reading indicates that

⁵⁰ Although to my knowledge no proponent of NA has argued for such a conception of lawful disunity, NA would nevertheless seem best placed to justify such a conception. For, on this conception, an irreducible plurality of independent laws could be grounded in an irreducible plurality of independent natures of things. Particular laws would be minimally, or distributively, unified insofar as they are all governed by the form of the a priori laws. But beyond this common character, the laws would have no further cohesion. ⁵¹ In a similar vein, Longuenesse (1998, 108) has described the universality of, e.g., causal relations as ‘immanent to empirical objects themselves.’ ⁵² This is suggested, moreover, by Kant’s conception of biology as informed by teleological principles. On laws in biology, see Breitenbach (2017). I believe that this non-reductive notion of unity leaves room for the conception, prevalent among a number of nineteenth-century scientists, that empirical generalizations are ultimately grounded on fundamental a priori principles. What it rules out is the further thought that all empirical laws are, in the end, reducible to one and the same set of principles. ⁵³ A654/B682. ⁵⁴ See A657f./B685f. ⁵⁵ Kant presents an organicist conception of the unity of nature in CJ 5:379. I argue for such a nonreductionist conception in Breitenbach (2009, chapter 6, and 2017).

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Kant’s notion of the lawful unity of nature is crucially distinct from any hierarchical or reductionist conception of the unity of nature.

5. Conclusion In this chapter, I have been concerned with Kant’s conception of empirical laws and with the unity of nature these laws generate. In particular, I have asked how on Kant’s account we can have knowledge of empirical laws, if the necessity of such laws is grounded in conditions that lie beyond our experience. On the reading I have suggested, in most cases we cannot know the empirical laws by derivation from these a priori principles, but have to discover them by reflection on the phenomena. Such reflection is an intellectual activity that centrally relies on the regulative idea of unity. Reflection of this kind cannot ground scientific knowledge in the strict sense of the term, but it can lead to a common form of knowledge, that is, more or less unified cognition. Although, on Kant’s account, the lawful unity of nature is thus ultimately inaccessible to us, it is nevertheless a proper aim of empirical inquiry.⁵⁶

⁵⁶ I thank the editors of this volume for helpful comments on an earlier version of this chapter. I am also grateful to Yoon Choi, Jim Kreines, Michela Massimi, and James Messina for stimulating discussion and constructive feedback. Last but not least, I gratefully acknowledge the support of the Riksbankens Jubileumsfond and the Swedish Collegium for Advanced Study.

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7 Becoming Humean John W. Carroll

1. Introduction Philosophers often seek an analysis of lawhood, a necessarily true completion of: P is a law of nature if and only if _____. There is much agreement that certain ways of filling in the blank are unsatisfactory. For example, analyses that use subjunctive conditionals are thought to do little to help us better understand what it is to be a law. Concerns about these conditionals are similar to, and just as challenging as, concerns about lawhood. The same can be said for filling in the blank with other nomic concepts like lawhood itself, causation, explanation, chance, dispositions, and their conceptual kin. As a result, many philosophers have shown a preference for a thoroughly non-nomic analysis of lawhood. Some of them are even more demanding: So-called Humeans attempt to analyze lawhood without using any nomic terms and without referring to any nomic ontology (e.g., law-making universals). In Laws of Nature,¹ I argue that Humeans have set themselves an impossible task. My core argument is the Mirror Argument, which provides an anti-supervenience example intended to show that a world’s laws need not be determined by its Humean facts. Others have given similar arguments.² Among them is an argument by Tim Maudlin that is distinctive by virtue of its appeal to scientific practice.³ Following this introductory section, the Mirror Argument and Maudlin’s Argument from Scientific Practice are briefly rehearsed in Section 2. These arguments purport to show that there are at least two possible worlds that agree on their Humean base, but differ on what their laws of nature are. That would be trouble for the Humeans because it would show that there is no necessarily true way to fill in the blank while respecting Humean scruples— any Humean analysis would be wrong about what the laws are in at least one of the two worlds. In ‘Nailed to Hume’s Cross?’,⁴ I defend my anti-Humean position against certain ¹ Carroll (1994, 60–8). ² Tooley’s anti-supervenience examples (1977, 669; 1987, 47–8, 67) were a major influence in my development of the Mirror Argument and its predecessor anti-supervenience examples given in Carroll (1990, 212–17). Other anti-supervenience cases are presented at least for discussion by Putnam (1978, 164), Earman (1986, 100), Menzies (1993, 199–200), and Lange (2000, 85–90). ³ Maudlin (2007, 67–8). ⁴ Carroll (2008, 73–5).

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metaphysical and epistemological concerns. I remain convinced that my responses to those challenges are sound. My defense includes a sketch of a nomic analysis of lawhood that identifies the laws as the regularities caused by nature. To be a bit more careful, the analysis states that P is a law of nature if and only if P is a regularity that is true because of nature. This is a nomic analysis since it uses the causal/explanatory term, ‘because.’ There are, however, other Humean issues that need attention. One suggests that the anti-supervenience arguments are unconvincing because they improperly presume that laws of nature govern. This manner of objection gained a foothold with Helen Beebee’s ‘The Non-Governing Conception of Laws of Nature.’⁵ Another objection claims that there is a subtle failure to recognize a change of context in the presentation of the anti-supervenience arguments. This challenge comes from John T. Roberts in his The Law-Governed Universe.⁶ The governing concern is discussed in Section 3. Digging into the idea of what it is for any kind of law to govern helps to relieve Beebee’s and other Humeans’ concerns that something illicit has been embraced by the anti-Humeans. The digging will also uncover Roberts’s sensible understanding of governing that, along with being sensible, is shown in Section 4 to fit well with my analysis of lawhood that treats laws as regularities caused by nature.⁷ Roberts’s concern about anti-supervenience is addressed in Sections 5–7. The concern is frustratingly compelling and suggests to me how to be Humean. So, alas, there is a new Humean story that needs telling. It includes a semantics for natural language modal sentences that contains an account of conditionals. The semantics is based on some mainstream ideas from linguistics (tweaked to eliminate the use of possible worlds). When this semantics is attached to my caused-by-nature analysis of lawhood, the result is a concept of lawhood that is rich in key ways: Laws of nature are not mere regularities, and they do govern. This concept of lawhood sits well with scientific practice. It is also suited to explain anti-Humean intuitions: With this conception of lawhood, given a single Humean base and a generalization P that is true of this base, it can be true to say, ‘P is a law,’ but it can also be true to say, ‘P is not a law.’ That would be significant anti-Humean desiderata met without contradicting Humeanism. Nevertheless, should this story be true (and I am not so sure it isn’t), the anti-supervenience arguments for anti-Humeanism would be based on a conflation.

2. Two Anti-Supervenience Arguments 2.1 The Mirror Argument Consider a possible world, U₁.⁸ In it, there are two X-particles—particles a and b—and two Y-fields (Figure 7.1). Since the beginning of time, these particles have been ⁵ Beebee (2000, especially 573, 580–1). ⁶ Roberts (2008, 357–61). ⁷ Roberts (2008, 46–8). ⁸ This is a much briefer (and less thorough) version of the Mirror Argument that is presented in Carroll (1994, 60–8).

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a

b

Figure 7.1

traveling in a line at a constant velocity toward the Y-fields. Each X-particle enters one Y-field. While in their respective Y-fields, both X-particles have spin up. Unlike particle b, particle a has an interesting mirror right along, though not in, its path to the Y-field. This mirror is on a swivel and so can easily be twisted. It is in fact in a position that does not interfere with a’s flight. If twisted, however, the mirror would deflect a out and away from all the fields. Clearly, L₁, the generalization that all X-particles subject to a Y-field have spin up, could be a law in such a world. Possible world, U₂, is just a little different. As in U₁, there are the two X-particles and two Y-fields. The X-particles again travel in a line, and each enters its Y-field at exactly the same time and place that it did in U₁. There is also that same twistable mirror. The only concrete differences in the histories of U₁ and U₂ involve what happens to a as it enters its Y-field. Specifically, in U₂, when a enters its field, it does not acquire spin up. Aside from this difference, there must be at least one decidedly nomic difference between U₁ and U₂: L₁ is not a law in U₂. Given my description of U₂, L₁ is not a law because it is not true. U₁ and U₂ are just two different ways our world could be. There is nothing remarkable about either. But here is the catch. It is natural to think that L₁’s status as a law in U₁ does not depend on what position the mirror is in. It is clear that if the mirror had been twisted then L₁ would still be a law of U₁. It is just as natural to think that L₁’s status as a non-law in U₂ also does not depend on the position of the mirror. L₁ would not be a law in U₂ even if the mirror had been twisted. All of this suggests that there are two more possible worlds—U1* and U2*—that we should be considering: U1* is the world that would result were the mirror twisted in U₁, and U2* is the world that would result were the mirror twisted in U₂. (So in both U1* and U2* particle a is deflected off its initial path and never enters any Y-field.) In U1*, L₁ is a law. In U2*, L₁ is accidentally true and so not a law. The question the Humeans face is how they are going to ground the fact that L₁ is a law in one of these worlds but not the other. U1* and U2* agree on their Humean bases but disagree whether L₁ is a law of nature. Therefore, U1* and U2* are a counterexample to supervenience.

2.2 The Argument from Scientific Practice Maudlin presses the case against the Humeans by focusing on the common practice among physicists (and other scientists) of considering models of a theory’s laws.⁹ ⁹ Maudlin (2007, 67–8).

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Let us suppose (and how can one deny it) that every model of a set of laws is a possible way for a world governed by those laws to be. Then we can ask: can two different sets of laws have models with the same physical state? Indeed, they can. Minkowski space-time, the space-time of Special Relativity, is a model of the field equations of General Relativity (in particular, it is a vacuum solution). So an empty Minkowski space-time is one way the world could be if it is governed by the laws of General Relativity. But is Minkowski space-time a model only of the General Relativistic laws? Of course not! One could, for example, postulate that Special Relativity is the complete and accurate account of space-time structure, and produce another theory of gravitation, which would still have the vacuum Minkowski space-time as a model. So under the assumption that no possible world can be governed by the laws of General Relativity and by a rival theory of gravity, the total physical state of the world cannot always determine the laws.¹⁰

The specific example Maudlin introduces here involves a matter-less universe with the laws of General Relativity and another with laws of a conflicting theory of gravitation. Despite the scientific practice, Humeans must contend that these pairs of so-called possibilities are not genuine possibilities. Maudlin summarily and seemingly reasonably recommends that we prefer ‘philosophical analyses that follow scientific practice to analyses that dictate it.’¹¹

3. Do Laws of Nature Govern? Beginning in the mid-1990s, a number of Humeans argued that the Mirror Argument and other arguments yielding anti-supervenience examples did not describe genuine possibilities. About the Mirror Argument, they held that the intuition about the lawfulness of L₁ in U1* or the intuition about L₁’s lack of lawfulness in U2* was somehow misleading or ill-founded. Many of these challenges turned on the issue of whether laws of nature govern and the accusation that anti-Humeans had adopted such a governing conception of laws. Here Beebee succinctly states the objection: The intuition that laws govern is, I think, deeply felt—at least implicitly—by lots of philosophers, and probably by a lot of the folk too. But it is an intuition that the Ramsey-Lewis view—and Humeanism about laws in general—refuses to endorse. And it is no accident that it refuses to do so: the intuition that laws govern is precisely the intuition which leads one to postulate the necessary connections—as ontological grounds of the governing nature of laws— which the Humean refuses to allow into her ontology.¹²

The idea is that, if one comes to the debate with the governing conception in mind, one is likely to find the Mirror Argument convincing, but using this conception to

¹⁰ Maudlin (2007, 67). ¹¹ Maudlin (2007, 68); also see Ismael (2015, 191). ¹² Beebee (2000, 573); also see Loewer (2004, 187–8), Roberts (1998, 436), and Schaffer (2008, 94–6).

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reject Humean analyses of lawhood is somehow to beg the question or to otherwise be unconvincing because it is a conception Humeans reject. Despite being an anti-Humean, I was reluctant to take seriously the idea that laws of nature govern. I did not see how laws could be the kind of entity that could do anything to the world. Laws (if they really are entities at all) are references of ‘that’ clauses. For example, it is a law that no signals travel faster than light. Traditionally, ‘that’ clauses are taken to refer to propositions, which are thought to be abstract entities, akin to numbers in their abstractness. I did not see (and still do not see) how laws any more than numbers could affect material objects and events. In Laws of Nature, I do talk about laws together with initial conditions governing or (more often) determining facts, but I thought of the laws as determining and governing as nothing more than a matter of logical entailment from the laws together with initial conditions. So it seemed to me that I was not guilty of presuming any governing conception of lawhood. Indeed, it was a conception I was inclined to reject. In something of the same spirit, Maudlin also employs the notion of governing, but only as part of ‘metaphors to fire the imagination.’¹³ Nevertheless, Roberts has an insightful discussion of governing that has changed my mind about the governing conception of laws.¹⁴ He describes what I call his sensible understanding of governing. Roberts points out that, even when we consider laws governing a nation, the laws do not do any pushing or pulling of the governed— the laws are not causal. What there is is a national government that creates and enforces the laws. ‘The proposition we call the law is not the agent of the governing, but the content of the governing.’¹⁵ This seems to me to be the correct way to think about how national laws govern. It is a way that can usefully be applied to laws of nature. How well do extant accounts of lawhood reflect the sensible understanding of what it is to govern? What I am looking for from these accounts is something that plays the role of the government. What is the author and the enforcer of the laws of nature? In line with a naive Humeanism, the simplistic answer would be that what goes on in the universe plays the role of the government. Anyone whose view of laws of nature accepts (i) that there are laws and (ii) that something is a law only if it is true, is in a position to hold that what goes on in the universe makes certain generalizations true. What goes on in the universe, then, can plausibly be taken to be the author of the laws of nature. There are, however, two problems here: First, what was just said applies to all true generalizations, the accidentally true ones as well as the laws. So this construal of how laws govern does not distinguish the governing of laws from the ‘governing’ of ¹³ Maudlin (2007, 15). Schneider (2007) defends the importance of the intuition that laws govern. She (2007, 317) and Loewer (2004, 196) recognize that Maudlin and I did not have any robust notion of governing in mind. Schneider (2007, 323) also endorses Loewer’s strategy of approaching antisupervenience and the governing intuition as just two issues in the debate between the Humeans and anti-Humeans; neither see the anti-supervenience examples as decisive on their own. ¹⁴ Roberts (2008, 45–8). ¹⁵ Roberts (2008, 46).

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any other true generalization. Second, nothing has been said about enforcement. According to the simplistic answer, there is nothing that prevents signals from moving faster than light. It is hard to find a plausible and satisfying analog for the government in the antiHumean literature. For David Armstrong and Michael Tooley, it would have to be the lawmaking universal that does the enforcing.¹⁶ For Maudlin, lawhood is a primitive concept; laws are primitive entities.¹⁷ His useful talk of governing stops at the level of metaphor. For John Foster, enforcement resides in the agency of God.¹⁸ I find none of these accounts illuminating regarding lawhood or governing. For Marc Lange, it is the counterfactual relations between the sub-nomic facts that pick out certain true generalizations as laws.¹⁹ It is not clear, however, what he might go on to say about governing. If he embraces a thick notion of governing, he might well think that it is the holding of these counterfactual relationships that is responsible for the governing. So the counterfactual facts plausibly become the authors of what the laws are and also arguably constitute what it is for the laws to be enforced. Plausible as this sounds, however, the role of the enforcer is left vacant as was the case with the simplistic answer. Constituting what it is for the laws to be enforced is not the same as enforcing the laws. I am looking for something that makes sure that those signals obey the cosmic speed limit. What about Roberts’s Humean view? Following Lange, Roberts employs a principle, NP, relating lawhood to counterfactual conditionals. It states (roughly) that Q is a law only if Q would still have held under any antecedent P that is consistent with the laws. Roberts gives NP a special role in his book: he holds that NP explicates the thesis that laws of nature govern²⁰—NP is what the lawful governing properly amounts to. What is not clear, however, is how this fits with Roberts’s sensible understanding.²¹ For him, the NP-sentence is context dependent. It is not true in all contexts. Roberts argues that it is true in the scientific contexts, and since it is true in those contexts, the sentence ‘Laws govern’ is also. Unfortunately, none of this reveals what it is that authors and enforces the laws. Could the enforcers be the facts expressed by the counterfactual sentences that are true in the scientific contexts? By his lights, yes, this would get ‘laws govern’ to be true in these contexts, but we seem no better off than we were with Lange’s view. At best these facts constitute the governing.

¹⁶ Armstrong (1983) and Michael Tooley (1977, 1987). ¹⁷ Maudlin (2007, 17–18). ¹⁸ Foster (2001, 147–9). ¹⁹ Lange (2009b, 15–20). ²⁰ Roberts (2008, 191–8). ²¹ In fairness to Roberts, he employs an analogy concerning the rules of English grammar to make the case that laws might govern without there being ‘a concrete agent that it makes sense to think of as the governing agent’ (2008, 47). Of course, I agree that there is no unique agent, concrete or otherwise, that enforces the rules of grammar, but I have known plenty of editors, journal referees, and school teachers who take enforcement very seriously. (The move away from the sensible understanding of governing takes hold as Roberts’s focus becomes laws being eternally and inevitably true; see his (2008, 174–6).)

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4. Laws Caused by Nature In Laws of Nature (1994),²² I left lawhood unanalyzed, only arguing for the negative conclusion that lawhood was not analyzable in purely Humean terms; and, like I said, I ducked questions about whether laws governed. As I also said, in a more recent paper, I give an analysis of what it is to be a law in terms of causation/explanation.²³ When I was writing the paper I was still not tuned into the question of whether laws govern; the word ‘govern’ does not appear in this paper. Nevertheless, the analysis described there fits the sensible understanding of governing better than most. Laws are not true as a matter of accident. They do not just happen to be true. Something accounts for their truth. That, however, is not all there is to being a law. For example, some particular states of affairs, like there being tobacco in North Carolina, are not a matter of happenstance but are not suitably general to be laws. For a more interesting case, it might be true that there are no gold spheres greater than a mile in diameter because there is not enough gold in the universe. In that case, strictly speaking, it would be true, suitably general, and not an accident that all gold spheres are less than a mile in diameter. Nevertheless, my considered judgment is that even in this case it still would not be a law; it is not enough to be a law to be general and not accidental. What seems important about this gold-spheres example is how the regularity turns out not to be accidental. It is non-accidental just because of the limited amount of gold. Something about what is in nature or is absent from nature makes the regularity true. Contrast this with the law that no signals travel faster than light. With this generalization, it seems that it is true because of nature. Lawhood requires that nature—understood as something distinct from anything in nature—makes the regularity true: P is a law of nature if P is a regularity caused by nature. While this is a catchy way to put my favored analysis of lawhood, there are three aspects of my view that require comment. First, philosophers will be uncomfortable with the idea of a regularity—a universal generalization—being an effect. That is OK. The key notion here is better expressed by ordinary uses of ‘because’. Strictly speaking, it is an explanatory notion. It is, however, different from causation in relatively uninteresting ways. If b’s being F caused c’s being G, then c was G because b was F. The other direction is not as straightforward: 3 is a square root of 9 because 3
3 is 9, but we are reluctant to say that 3
3’s being 9 caused 3 to be a square root of 9. We are reluctant to take mathematical explanations, explanations underwritten by definitions, and explanations of or by universal generalizations to be causal ones. For some reason, when the explanandum and the explanans are too closely connected, connected in some more-orless analytic way, and not by some paradigmatically causal process (e.g., colliding), or when the explanandum or explanans themselves are sufficiently unlike paradigmatic ²² Carroll (1994).

²³ Carroll (2008).

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causes and effects (e.g., a spark or an explosion), philosophers tend not to consider the explanations to be causal. That, however, seems to me to be the bulk of the difference. In any case, officially, my view is that P is a law of nature if and only if P is a regularity that is true because of nature. Second, self-respecting metaphysicians will surely ask what exactly nature is. Think of nature as the universe itself—not the objects and events in the universe, but whatever it is that the objects and events are in. Along this same line, we can think of nature as something like the universe’s space-time manifold or the totality of its space and time. Better yet, think of nature as something like an omnipresent and eternal field, a big-as-big-can-be magnetic field that is also as long-lasting as longlasting can be, whose effects need not have anything to do with magnetism. Some will object to the idea that something like nature or the universe itself can make certain regularities true. Nature is not an event. It is also not a state of affairs (i.e., an object or event having some property). Yet, it is the opinion of many metaphysicians that only events or states of affairs can cause anything. To some it may even sound like I am taking seriously the idea of substance causation, an idea that is often in disrepute. Nature is not a substance, exactly. It is more like a field; it contains substances, but is not itself one. Taking nature to be causal is no more worrisome than thinking of a magnetic field as causing an electron to move in a certain manner. Furthermore, my account leaves room for states of affairs to play a role. Nature causes and explains what it does by virtue of being the way it is. Just so, it is useful to formulate the caused-by-nature analysis thus: P is a law of nature if and only if P is a regularity that is true because nature is the way it is. Third, now that we have a better idea what nature is, let us ask what it is for nature to be a certain way. Is my being in nature a way nature is? Is the amount of gold in nature a way that nature is? Yes, these are two ways nature is, but that is not what I have in mind when I discuss the way nature is. This phrase is intended to pick out the one intrinsic (but possibly multifaceted) way the universe itself is. Here is a helpful, somewhat humble, analogy: Think of a cafeteria tray that we do not know much about. We do not know its age, composition, whether it has structural damage, and so on. If the tray breaks under the weight of the food even though I was carrying it carefully with both hands, then it is likely true that the food fell on the floor because of the way the tray was. This is not a particularly illuminating explanation, but it is a true explanation. Indeed, the phrase ‘the way nature is’ belongs in the analysis of lawhood because of its limited informativeness and for its being obviously true that nature is the way it is. The obviousness of the judgment that nature is the way it is and the judgment’s nearly total lack of detailed information sustains the scientific challenge to understand better what that way is like. This challenge reflects exactly the situation that scientists face as they work to reveal what are the laws of nature. There is no other way around. On the caused-by-nature analysis, by encountering many failures to falsify a generalization P, it may become reasonable to conclude that P is true because of

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the way nature is. In this manner, stating that P is a law is science’s approach to indirectly describe how nature is by virtue of which it causes P. That laws of nature govern is underwritten by the analysis, which puts nature in the role of the government. The enforcement is part and parcel of the legislative process as it is a causal matter of nature making the basic particles and everything else in the universe behave as they do given the properties that they have. As a result, they follow the rules. Nature is what prevents signals from traveling faster than the speed of light.

5. Roberts on Anti-Supervenience Roberts offers a new manner of responding to apparent counterexamples to supervenience.²⁴ Here is a simple such counterexample: For one with anti-Humean leanings, it will seem that there are two lonesome-particle possible worlds. In one of these worlds, it is true that there exists only a single particle traveling at constant velocity throughout all of history and it is a law that F ¼ ma. In other worlds, it is true that there exists only a single particle traveling at constant velocity throughout all of history and it is not a law that F ¼ ma. For Roberts, this reasoning goes wrong because, though ‘It is a law that F ¼ ma’ is true relative to a context/world pair and may also be false relative to a different context/world pair, that is not enough to challenge supervenience. The trouble is that the difference in truth value between the two sentences at the respective worlds could be determined by differences in the contexts and not by differences in the worlds. This leaves open that there is really only one lonesome-particle world. Without going into much detail about Roberts’s metatheoretic account of lawhood, he holds that for a possible world w in which there exists only a single particle traveling at constant velocity throughout all of history and relative to a context in which the salient theory is, say, Newtonian Mechanics, ‘It is a law that F ¼ ma’ is true just in case F ¼ ma plays the law role in the salient theory and this theory is true in w. As Roberts sees it, F ¼ ma cannot both play the law role and not play the law role relative to a single theory, and so a different salient theory and thus a different context is required for ‘It is not a law that F ¼ ma’ to be true relative to any other world in which F ¼ ma is true. Our ‘two lonesome-particle worlds’ which seemed to pose a counterexample to HS [Humean Supervenience] are not two distinct possible worlds; they are one possible world, considered from two different points of view—one possible world referred to in two different contexts of utterance.²⁵

Roberts has shown how it can seem that there are two worlds with the same Humean base and different laws, even though there is only one world such that relative to two different contexts a single lawhood ascription has different truth values. ²⁴ Roberts (2008, 357–61).

²⁵ Roberts (2008, 359).

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What is unique (and for me, uncomfortably compelling) about this reply is that it does not involve saying that any claim about laws and possible worlds that is supported by our intuitions and mobilized in the anti-HS argument is false.²⁶

About my Mirror Argument, given the Humean base stipulated for both U1* and U2*, Roberts thinks it is true to say, ‘L₁ is a law’ in certain contexts about U1* and true to say, ‘L₁ is not a law’ in certain other contexts about U2*. But, according to Roberts, my mistake was not recognizing that, for all that was just said, U1* may be U2*. The antiHumean judgments about what are the laws are reasonable judgments, but—by his lights—there was also a failure to recognize the influence of context. Parallel points apply to the Argument from Scientific Practice: Maudlin’s so-called two possibilities would be seen by Roberts as descriptions of a single possibility that are made relative to two contexts with different salient theories: General Relativity and some rival theory of gravity. What I do not find compelling about Roberts’s position is his view on the context dependence of lawhood ascriptions. His view is devised for one particular phrase of English: ‘law of nature.’ For lawhood ascriptions to be true relative to a context, there must be a theory that is salient in the context. Here is the beginning of his final characterization of what being the salient theory involves: MT₃: The default salient theory in a context k is that theory which comprises all the true theoretical commitments of the members of the extended community of speakers.²⁷ Roberts goes on for several more lines in his full statement of MT₃, describing how the default might be cancelled; and he acknowledges that it is ‘quite a mouthful.’ Prima facie, I take this complexity to be suspect as part of a characterization of what is supposed to be a component of conversational contexts. Even just focusing on the beginning portion of MT₃ stated above, the sophisticated notion of salient theory tied to the theoretical commitments of speakers seems out of place. Some added complexity might be expected in a consideration of ‘law of nature’ as it is at least a semiscientific phrase. It is not as common or central in ordinary conversations as terms like ‘cause,’ ‘possible,’ and ‘if . . . then —’ for which such complexity would seem even more problematic. Still, it would be better if the contextual treatment of ‘law of nature’ melded neatly with the context dependence of other natural language words and phrases, especially the nomic words and phrases. ‘Law of nature’ should not be an isolated freak of our language.²⁸ To put the point differently, Roberts’s contextual treatment of ‘law of nature’ feels ad hoc. We should try to understand the context dependence of our nomic language by appeal to linguistic principles, and the investigation should be driven by considering conversational practice. It should not be driven by the preferred formulation of ²⁶ Roberts (2008, 360). ²⁷ Roberts (2008, 117). ²⁸ Cf. Unger (1971, 202) on the verb ‘to know.’

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philosophical theories from metaphysics or the philosophy of science. In the remainder of the chapter, I will sketch a more realistic story about the context-dependence of modal terms that I see as a more plausible way to implement Roberts’s challenge. It is not the only way, but it is the way that is threatening to make me Humean.

6. Contextual Semantics Here is a new contextual (sentence) semantics for natural language modal terms applied to ‘possible’ (◊), ‘necessary’ (⬜), and ‘if . . . then —’ ( . . . > —).

6.1 Formal Semantics The script letters ‘P,’ ‘Q,’ and ‘R’ are metavariables for sentences. ⬜ and ◊ are the modal operators; > is a modal connective. The syntax is just what you would expect. For my purposes, there is no need to introduce predicates or quantifiers. A model is a context, L, together with a truth assignment to the atomic sentences and to the truthfunctional compounds (with ~, v, &, or ⊃) in the usual way. L is a set of sentences. Then, here are the truth conditions for ⬜, ◊, and >: ⬜P is true in L iff P is true and a member of L. ◊P is true in L iff ~⬜~P is true in L. P > Q is true in L if and only if ⬜(P ⊃ Q) is true in L.

This semantics is not truth functional. The truth values of some compound sentences are not determined by truth values of their components. For example, if P is true and in L, but Q is true and not in L, then ⬜P is true and ⬜Q is false, even though P and Q are both true.

6.2 Interpretation ⬜ is intended to represent natural language uses of necessity modals like ‘must,’ and ‘is necessary,’ ◊ represents such uses of possibility modals like ‘might’ and ‘is possible,’ and > such uses of ‘if . . . then —.’ Think of L as a portion of David Lewis’s scoreboard in ‘Scorekeeping in a Language Game,’²⁹ a portion crucial to the truth valuation of natural language modal sentences. An assertive utterance of a sentence in a conversation takes place against an informational background, a set of propositions, roughly, the propositions that have been supposed, presupposed, taken as settled, taken as granted, the presumed to be shared information. This background is what Robert Stalnaker calls the common ground. Be aware that the background changes over the course of an ordinary conversation. ‘Assertions . . . are proposals to change the context by adding

²⁹ Lewis (1979b).

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the information that is their content.’³⁰ With an assertive utterance of a sentence, the proposition expressed becomes a candidate for inclusion in the common ground. Here is a simple illustration. Suppose we are inside a big building, and do not hear any rain drops, but there was a report of a 20 per cent chance of rain. We are deciding whether to take our lunch outside. If I say, ‘It is possible that it is raining,’ then I say something true, because that it is not raining is not common ground—that ~P is not in L is sufficient for ◊P. Moreover, it is certainly not settled that it is raining—that is not common ground. Prima facie, if I say, ‘It must be raining’ or ‘It is necessary that it is raining,’ it seems that I have said something false. This is just what the semantics reports, because in the conversation it is not common ground that it is raining. What of >? According to the formal semantics, P > Q is just a specific kind of necessity sentence. Indeed, it is modeled on the traditional notion of the strict conditional.³¹ On the intended interpretation, ‘If P then Q’ is true if and only if that P implies Q is both true and common ground.³² ‘>’ is intended to represent natural language uses of ‘if . . . then—’ including both subjunctive and indicative conditionals. Here is the idea: the mood and/or tense of a sentence indicates not a difference in the semantics of two different conditionals; instead, it provides information that shapes the background information prior to the truth valuation of the utterance. Consider a minor variation of the well-worn Oswald/Kennedy example.³³ Such examples are often used to illustrate a difference between indicative and subjunctive conditionals. (1) If Shakespeare did not write Hamlet, then someone else did. (2) If Shakespeare had not written Hamlet, then someone else would have. With (1), being sensitive to the words chosen by the speaker, we take it that the speaker has some doubt or someone else has raised some doubt about whether Shakespeare wrote Hamlet. We are inclined to think that the speaker does not take it as settled that Shakespeare did write Hamlet, because the speaker has chosen not to use the subjunctive. Either because we share those doubts or just go along with the doubts to keep the conversation moving along smoothly, that Shakespeare wrote Hamlet is not likely to be common ground. Still, we have no reason to think that the speaker (or anyone else) doubts that Hamlet was written by someone. It fact, that matter is pretty much common knowledge and so is likely common ground.

³⁰ Stalnaker (1999, 111). ³¹ Similar approaches have been taken up by Brogaard and Salerno (2008), Gillies (2007), von Fintel (2001), and Warmbrōd (1981). For an overview of recent discussions of conditionals, see von Fintel (2012). ³² In most instances, I will use ‘P implies Q’ as a natural language counterpart for the material conditional, ‘P ⊃ Q.’ Where it is more perspicuous, I will instead use the standard disjunctive equivalent: ‘not P or Q.’ ³³ Von Fintel (2012, 466).

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Is (1) true with this common ground? Is it settled that either Shakespeare wrote Hamlet or someone else did? Well, the rule of accommodation (and the standard purpose of pairs of examples like these) certainly urges it to be. And why not? It follows from the piece of common knowledge that someone wrote Hamlet that either Shakespeare did or someone else did. That is why the common judgment about (1) is that it is true. With (2), that Shakespeare wrote Hamlet is naturally assumed to be plausible to the speaker. The speaker’s use of the subjunctive indicates the antecedent’s remoteness from the actual, either indicating that the antecedent is thought to be outright false or at least that there might be doubts about the antecedent. This (and that who is Hamlet’s author is pretty well known itself) suggests that the speaker knows that Shakespeare did write Hamlet. So, it is somewhat natural initially to take that Shakespeare wrote Hamlet as being part of the common ground. But it is not likely to stay there. This is something that needs to be jettisoned from the common ground for a proper consideration of the conditional. To avoid triviality, we should shift the common ground so that, relative to the new context, it is not common ground that Shakespeare wrote Hamlet and not common ground that someone else did.³⁴ It is likely to be part of the original common ground and continue to be in the new common ground that no one other than Shakespeare was gifted or devoted to theater enough to write a masterpiece like Hamlet. If so, we are likely not to be prepared to take as settled that either Shakespeare wrote Hamlet or someone else did. Since this common ground includes that no one other than Shakespeare was gifted or devoted to theater enough to write a masterpiece like Hamlet, it seems reasonable not to include that Shakespeare did not implies someone else did (i.e., either Shakespeare did or someone else did). So, relative to this common ground, (2) is false. Be careful. The point of considering (1) and (2) might mistakenly be thought to be an illustration of the distinct truth conditions for indicative and subjunctive conditionals. With the intended interpretation of my semantics, the example does no such thing. The reported difference in the truth values of utterances of (1) and (2) are due to the differences in the contexts for truth valuation generated by the mood/tense of the antecedents and consequents. The semantics includes only the one conditional.

³⁴ Von Fintel (2001, 134–5) reports that Irene Heim made the proposal that, roughly, subjunctive conditionals presuppose that their antecedent is possible. Von Fintel adopts Heim’s proposal. I do, too, but my semantics suggests that there is more going on. The presupposition that the antecedent is possible— and so its negation is not common ground—in part arises from triviality concerns: If that Shakespeare did write Hamlet is common ground, then it trivially follows that Shakespeare did write Hamlet or someone else did, and so that is likely common ground too, in which case (2) would be rather trivially true. For pretty much the same reason, we should expect there to also be some pressure to presuppose that the consequent is possibly false—so the consequent should not be in the common ground. If it is common ground that someone else wrote Hamlet, then (2) is again rather trivially true.

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6.3 Kratzer Semantics I hope I have said enough for the reader to have a feel for how the semantics works. If not, I am fortunate that the semantics is similar to some more mainstream work in linguistics, especially Angelika Kratzer’s influential work.³⁵ Cast within a standard possible-worlds framework, her approach has the conversational context determine a set of accessible worlds. Possibility is tied to truth in at least one accessible world; necessity is tied to truth in all accessible worlds. The modal base is a set of propositions that determines which possible worlds are the accessible ones. The accessible worlds are exactly the worlds for which all the members of the modal base are true. It has become standard fare in the linguistics community to think of the modal base as given by the common ground. So, my L serves in a manner that is similar to the role of the modal base in a Kratzer semantics. Indeed, if a proposition P is true and a member of the modal base, then ⬜P, just as: if P is true and P is in L, then ⬜P. Though I do not do so here, one could implement Roberts’s challenge to the antisupervenience examples with a Kratzer-style semantics that treated ‘if . . . then —’ as the strict conditional.³⁶

7. A Way to be Humean Time to see how a governing conception of laws can be consistent with Humeanism, but it is also time to see how the caused-by-nature analysis of lawhood is not threatened by this way to be Humean. To get a handle on the context dependence of ‘because,’ I introduce a conditional treatment of ‘because’ that holds: Q because P if and only if, if P were the case, then Q would be the case. It is important that the analysis should be understood as requiring the true presupposition of both P and Q; that is a plausible semantic feature of ‘because’ sentences and warrants treating the conditional in the analysis as in the subjunctive mood. To seamlessly attach this analysis to the caused-by-nature account, let us take the caused-by-nature analysis to hold that P is a law of nature if and only if P is a regularity that is true because nature is the way it is.³⁷ ³⁵ Much of which is included in Kratzer (2012). ³⁶ There are at least four important differences between my semantics and the Kratzer semantics. First, my semantics does not treat propositions as sets of possible worlds. For the purposes of this chapter, I take propositions to be primitive entities. Second, there are metaphysically necessary propositions that Kratzer’s semantics counts as necessary in all contexts, but my approach allows contexts such that even a metaphysically necessary truth P be such that ◊~P is true; this would be the case if P were not in L—so, for example, for some contexts, ‘Goldbach’s Conjecture might be true and might be false’ is true. The third important difference is the work done by the membership relation in my semantics. Set membership allows a simple, plausible story about how modals help shape conversations. Fourth, Kratzer does not treat English ‘if-then’ sentences as strict conditionals; instead ‘if ’-clauses restrict the domains of certain operators (cf. von Fintel 2012, 471–2). ³⁷ There are potential problems for this conditional analysis. The most serious are provided by cases like Kyburg’s (1965) case of the hexed salt, but keeping the analysis simple is helpful for the purposes of

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Now, having just introduced the contextual semantics to the caused-by-nature analysis of what it is to be a law, there is the danger that some will think that I am stuck with the consequence that what it is to be a law of nature somehow is sensitive to what is going on in conversational contexts, on what the participants in a conversation are willing to agree upon or take as granted. Wouldn’t there be laws of nature even if there were no people, no conversations, and no contexts? Of course, there would be, but nothing about my semantics or the caused-by-nature analysis suggests otherwise. Basically, the semantics tells us that the truth of a lawhood sentence or utterance depends on there being a context for truth valuation reflecting a kind of agreement in virtue of there being an appropriate common ground. Something similar goes for standard semantics for sentences with pronouns like ‘he.’ For the sentence, ‘He is over six feet tall’ to be true, there needs to be a kind of agreement on what ‘he’ refers to in order for an utterance of that sentence to be true. Still, that he is over six feet tall does not depend on whether anyone agrees on anything. Similarly, even though the truth of the sentence, ‘It is a law that no signals travel faster than light’ does depend on certain features of a context, that it is a law that no signals travel faster than light does not. Turning back to Roberts’s challenge, focus again on a lone-particle Humean base. For F ¼ ma to be a law, F ¼ ma must be true because nature is the way it is. With our conditional treatment of ‘because’ that requires that, if nature were the way that it is, then F would still be equal to ma. For this conditional sentence to be true, it must be common ground that nature’s being the way it is implies F ¼ ma. If this is common ground, and since F ¼ ma is suitably general and true about the lonesome particle base, the sentence ‘F ¼ ma is a law’ is true. Given that F ¼ ma is suitably general and true about the lonesome-particle case, the only way this regularity could fail to be a law about this case would be if there was a different common ground, and hence a different context. It would have to be a context for which it is not common ground that nature’s being the way it is implies that F ¼ ma. There would have to be two contexts, but not two possibilities. The Mirror Argument is in similar trouble. In stating L₁ to be a law of U₁ we are committing to L₁ being true because nature is the way it is, and what comes with that is its being common ground that nature’s being the way it is implies L₁. Quite naturally, that continues to be common ground even as we suppose that the mirror has been twisted in U1*. So, since L₁ is suitably general and true about U1*, it is true to say, ‘L₁ is a law of nature.’ The story is only a little different with regard to the progression from U₂ to U2*. In stating L₁ not to be a law of U₂, we do so because L₁ is

understanding my implementation of Roberts’s challenge to the anti-supervenience examples. We might improve the analysis by adding another conjunct to the analysans such that, if P were false, then Q would be false (or maybe that it is not the case that if P were false, then Q would be true) to handle Kyburg’s case, but it will not make any difference regarding Roberts’s challenge. The anti-supervenience examples would still require a change of context.

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false about U₂. As a result, there is no reason to take it as settled that nature’s being the way it is implies L₁. In shifting our consideration to U2* with the supposition that the mirror has been twisted, we find that L₁ is true about U2*, but there is no background information provided by considerations about U₂ to prompt it to become common ground about U2* that nature’s being the way it is implies L₁. So, it is natural to conclude that L₁ is not a law of nature. Therefore, because of the required difference in the contexts, there is nothing here to suggest that U1* and U2* are not just one world. What of Maudlin’s Argument from Scientific Practice? The Humean should deny Maudlin’s opening assumption. Not every model of a set of laws is a possible way for a world governed by those laws to be. Every model of a set of laws is a possibility such that relative to certain contexts it can accurately be described as having the laws that are in the set. Every model of a set of laws is also a possibility such that relative to certain contexts it can accurately be described as not having the laws in the set, but as having other laws or no laws at all. Though my account captures reasonably well what is going on linguistically with this aspect of scientific practice, the account also shows that Maudlin’s argument does not establish anti-supervenience. There is nothing in the scientific practice to establish that there are two possible worlds that agree on their Humean facts, but have different laws.

8. Conclusion I am not quite ready to sign off on my anti-Humeanism. For the insightful, and maybe even for the not-so-insightful reader, I suspect that some of the limitations of my Humean story are glaring. Perhaps the most glaring is the newness and radicalness of the modal semantics introduced here in print for the first time. The other glaring limitation concerns the conditional treatment of ‘because.’ Conditional analyses of explanation and causation have been around for a long time and so have many apparent counterexamples. Even though the specific conditional analysis of the ‘because’ connective that I have used for illustration almost certainly needs to be sophisticated, I do think that a closer look at natural language and the role of context might provide new resources for addressing the apparent counterexamples. On behalf of other anti-Humeans, it is important to point out that some of the significant consequences of the anti-Humean project survive Roberts’s challenge and my suggested way to be Humean. For example, the impossibility of giving a thoroughly non-nomic analysis of lawhood that I defended using the Mirror Argument still strikes me as plausible even if the Mirror Argument is unsound. The considerations raised here go no measure to showing the possibility of there being an analysis of lawhood using only non-nomic terms. Even though a Humean might embrace the simplicity of my semantics, and the non-modal nature of the intended interpretation, I do not see how the semantics could help one give a non-nomic analysis of lawhood.

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The semantics give the truth conditions for certain natural language sentences—the semantics is not a conceptual analysis. If something like the Humean story I have told is, in fact, true, then one thing that I take away from the story is that the absence of a thoroughly non-nomic analysis of lawhood does not reflect the presence of the sui generis nomic modality that the Humeans—maybe fairly—accused anti-Humeans of embracing. In fact, it seems much less clear that a nomic analysis should be seen as unsatisfactory in any way at all given something like my semantics for possibility, necessity, and the conditional. The appearance of some distinctive modality would just be an offshoot of how our modal language works.³⁸

³⁸ Much of this chapter derives—sometimes pretty closely—from passages of my published and unpublished work. Marc Lange, John Roberts, and the other members of the Triangle Ellipse reading group provided useful comments and questions during our session on whether laws govern. John also commented on a near final version of this chapter. Walter Ott delivered comments on an early draft, reflecting thoughts from Walter, Lydia Patton, and students in Walter’s graduate seminar on laws of nature at the University of Virginia (especially Jim Darcy and Mason Pilcher). Ann Rives helped to finalize the presentation. Thank you to you all (and to others I might have missed) for your excellent support.

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8 A Perspectivalist Better Best System Account of Lawhood Michela Massimi

1. Introduction Two questions have catalyzed the debate surrounding laws of nature. Do laws govern nature? And what is the fundamental ontology of nature that is compatible with laws and their (governing or not) role? David Lewis has laid down an influential approach to these two questions. On Lewis’s account, laws do not govern nature because nomic facts reduce to non-nomic facts about natural properties. Humean supervenience (a tenet of Lewis’s account) maintains that modal facts supervene on this mosaic of non-modal facts about sparse natural properties. Lewis’s Humean ontology is modest: it consists of a (physics-driven) vocabulary of sparse natural properties (e.g. mass, charge, spin, among others), whose constant co-occurrence defines all there is. Thus, an electron is a well-defined cell in the Humean mosaic of co-occurrent natural properties (negative electric charge, half-integral spin, mass of 0.511 MeV/c²). Interestingly, some of these natural properties co-occur in more than one welldefined cell of the mosaic. For example, half-integral spin co-occurs with the property of negative electric charge (in the cell of properties defining the kind electron). But it also co-occurs with the property positive electric charge (in the cell of properties defining the kind positron). That is what makes Humean ontology a Legoland of possible reconfigurations of the same fundamental natural bricks (with no underlying causal glue), upon which natural kinds are mapped. Nomic facts about natural kinds are then said to supervene on this Legoland of natural properties. Pauli’s principle or Coloumb’s law equally supervene on nonnomic facts about the electron’s natural properties (e.g. the half-integral spin going hand in hand with antisymmetric states; the negative electric charge going hand in hand with repulsion of similarly negative electric charges, without any causal glue underpinning their going hand in hand). Lewis’s Best System Account

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(BSA) of lawhood¹ finds its ultimate justification in this neat ontological scenario: how nomic facts supervene on non-nomic facts about natural properties. For it contends that laws (do not govern but) simply are regularities; and for a regularity to be a law, it ought to fit with other laws in a system that achieves the best balance of simplicity and strength (the latter understood as information content). There are regular, co-occurrent, non-nomic properties about the electron’s spin, its electric charge, and a bunch of other natural properties that undergird Pauli’s principle and Coloumb’s law (among many others), and make them part of a best system. But no similar regular co-occurrence of properties is found to undergird, say, ‘All fruits in Smith’s garden are apples.’ The latter is not part of any best system. The goal of this chapter is to defend a suitably modified BSA account of laws, which is still kosher to the Humean spirit of Lewis’s proposal, but does not fall prey to some classical objections to Lewis. The modified version of BSA that I am going to propose deals in particular with the problems of subjectivity and nomic necessity that Lewis’s BSA typically faces. Under the modified BSA account suggested in this chapter, it is not the Humean mosaic of sparse natural properties alone that ultimately grounds laws of nature. But it is the Humean mosaic together with our standards of simplicity, strength, and balance that ground laws of nature. More to the point, I am going to argue that our standards of simplicity, strength, and balance change over time and are perspectival, in some relevant sense here to be clarified. I will be arguing that simplicity, strength, and balance are not standards pertaining to some non-better specified God’s-eye view (or what Ned Hall has aptly called Lewis’s LOPP, Limited Oracular Perfect Physicist).² Rather, they are perspectival standards adopted by real scientific communities across the history of science. Thus, my objective is to unpack some unexpected resources available to a Lewisian account of lawhood (when dealing with classical objections about subjectivity and nomic necessity), at the condition of rethinking the Lewisian tenet that the standards defining the best system belong to some ideal LOPP. As a consequence of this move, for a regularity to be a law, it has to feature within a perspectival series of Best Systems, defined by our historically evolving standards of simplicity, strength, and balance. Thus, the main point of my chapter is simply to reject the Lewisian idea that standards defining the Best System are fixed once and for all, as opposed to belonging to real historical communities that have interpreted those very same standards in very different ways across time. This is after all good news for Lewis’s view. Far from shaking the foundations of Lewis’s Best System, my perspectival move offers a way of reassessing the resilience of Humean regularities that we count as bona fide laws across the history of science and theory change. My start-up problem is the maybe unsurprising observation that our standards of simplicity, strength, and balance seem to have evolved remarkably across the history

¹ Lewis (1973, 73–4).

² See Hall (2015).

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of science. As a result, there seems to be more than one Best System featuring some important regularities that are non-controversially regarded as fundamental laws. Let us consider two paradigmatic examples from physics: the law of conservation of mass and the law of conservation of momentum. Mass is a natural property of fundamental particles par excellence. All fundamental particles can be classified on the basis of their identifying masses. New particles are discovered by identifying unexpected mass values against a background of known events (e.g. the 125 GeV of the Higgs boson; or the 3.1 GeV bump for the J/psi particle). What better candidate for a Lewisian natural property acting as the basis on which laws of nature (indeed, fundamental ones such as conservation of mass) supervene? Under the Lewisian story, one ought to say that conservation of mass (together with momentum conservation, and others) form a best system of nomic facts that maximizes simplicity in premises and strength (or information content) in the conclusions (where simplicity and strength are taken as fixed standards belonging to a LOPP). In other words, it is the existence of mass as a natural property of fundamental particles, and its co-occurrence in a Humean mosaic of non-nomic facts about other natural properties that underpin the law of conservation of mass (as well as other laws of nature) in our best system. Leaving aside here van Fraassen’s objection against Lewis’s ‘eschatology of science,’³ let us ask a slightly different (albeit related) question. Assuming for the sake of Lewis’s argument that scientists do try to achieve the best balance between simplicity and strength as their knowledge of the natural world increases, how many Best Systems have there been in the history of science where conservation of mass has featured as a ‘law’ (and, indeed, a fundamental one)? How do the Lewisian standards of simplicity and strength have to be understood in these different Best Systems? Let us go back to Antoine Lavoisier, who is usually credited for having discovered it. He certainly was not concerned with producing a Best System of axiomatized knowledge, from which conservation of mass could follow. If anything, Lavoisier’s gravimetric methods and his use of the balance for establishing the principle of conservation of weight (note: weight, not mass) has been rightly associated with his double career as a tax collector, given that at the time the balance was the instrument of assayers and apothecaries, more than physicists or chemists.⁴ More to the point, even granted that Lavoisier’s principle of conservation of weight might have featured somehow in a Lewisian Best System combining simplicity and strength, how did Lavoisier and his contemporaries understand simplicity? Simplicity and strength are notoriously vague notions. But when it comes to chemistry, it might be tempting to think of Lavoisier’s system of chemistry as being simple in the classification of chemical substances and in the treatment of chemical

³ van Fraassen (1989, ch. 3).

⁴ See Bensaude-Vincent (1992).

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compounds. The simplicity of Lavoisier’s system can be understood as the outcome of applying gravimetric methods to chemical compounds (e.g. weighing in and out every substance involved in chemical reactions) so that it was possible to reach conclusions about both the mosaic of non-nomic facts (e.g. that oxygen is a chemical element) and nomic claims supervenient on them (e.g. about oxidation of metals). Stahl’s chemistry at the time formed a different system, with different non-nomic facts (oxygen was not in the mosaic, but phlogiston was) and, as a result, different nomic claims too (about calcination and combustion). Its simplicity was not defined by the application of gravimetric methods to chemical compounds but by analogies and chemical prototypes.⁵ Was Lavoisier’s system better than Stahl’s? Yes, insofar as reliably identifiable regularities (thanks to gravimetric methods) are simpler than regularities identifiable via mere analogies and intuitions about what might count as a prototype in chemistry. In another sense, no, because when it comes to strength Lavoisier’s system allowed in non-nomic facts about caloric, as much as Stahl’s system slipped in nonnomic facts about phlogiston. Thus, Lavoisier’s system of chemistry, while regarded the best at its own time, did not exactly achieve the Lewisian LOPP-ian optimal balance of simplicity in the premises (given caloric still featured in it) and strength in the conclusion (given that caloric-related phenomena were meant to derive from it).⁶ In the age of Enlightenment, the regularity called ‘the law of conservation of mass’ was downstream of Lavoisier’s standards of simplicity and strength. But, historically, there have been other Lewisian Best Systems featuring mass conservation as a fundamental law. After all, it was the great success of Newtonian mechanics that placed mass center stage as an intrinsic natural property—key to an entire branch called point mechanics. In Newtonian mechanics, any body can be thought of as a mass point with mass m, positioned along Euclidean coordinates x, y, ! z, and velocity v (along the Euclidean coordinates). By combining mass and velocity, a new physical quantity can be obtained, momentum, and momentum is itself a conserved quantity—what was known to the Cartesians of the early eighteenth century as dead force (mv), as opposed to Leibniz’s living force 12 mv2 , which would later become known as kinetic energy and feature in the law of conservation of energy Z x 1 mv2  FðxÞdx ¼ const 2 0 (where the first term is what we now call kinetic energy and the second term is potential energy).

⁵ For the importance of gravimetric methods in Lavoisier vis-à-vis Stahl, see Gough (1988, 17–20). ⁶ ‘These discoveries give reason to hope that chemistry may one day arrive at the most beautiful state of simplicity. It is perhaps no improbable conjecture that all the bodies in nature may be referred to one class of simple combustible elementary substances, to oxygen and to caloric; and that from the various combinations of these with each other, all the variety produced by nature and art may arise’ (Lavoisier 1799, 224).

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It was one of the greatest achievements of classical mechanics that conservation of energy so defined could be deduced by integration from Newton’s second law of motion whenever the force function F is known. Moreover, it was one of the great mathematical achievements of the eighteenth century to show how from Newton’s laws it is also possible to derive D’Alembert’s principle and Lagrange equations. What would count as simplicity and strength in the Newtonian Best System where mass conservation (and momentum conservation) feature? Roger Cotes, in the preface to the second edition of the Principia, paradigmatically explained the kind of simplicity at work in the Newtonian system.⁷ Newton himself clarified what he meant by simplicity in his first rule of reasoning: i.e. ‘no more causes of natural things should be admitted than are both true and sufficient to explain their phenomena.’⁸ We have long abandoned this Newtonian way of interpreting simplicity as identifying true and sufficient physical causes for given phenomena. For example, Cartan’s geometrized gravitation theory no longer interprets gravitational force as a physical ‘cause’ for planetary motion because it endorses a variably curved inertial structure to define freely falling trajectories for bodies (similar to general relativity). And it has even been argued that simplicity should be understood along the lines of geometrized Newtonian gravitation with its curved spacetime as the best natural setting for Newton’s gravity (a setting that Newton himself would have preferred, had he known the mathematics for it).⁹ More to the point, already in the early nineteenth century, Hamilton came up with an alternative way of systematizing motions of mass point bodies, which instead of force laws, starts with a Lagrangian function featuring in what became known as Hamilton’s principle. And Hamilton’s principle redefined simplicity not in terms of deriving the ‘causes of things from the most simple principles’ (i.e. force laws), but in

⁷ ‘Those who fetch from hypotheses the foundations on which they build their speculations, may form indeed an ingenious romance, but a romance it will still be. There is left then the third class, which professes experimental philosophy. These indeed derive the causes of all things from the most simple principles possible; but then they assume nothing as a principle that is not proved by phenomena. They frame no hypotheses’ (R. Cotes’s Preface to the Second Edition of Newton’s Principia in Newton 1687/1934). ⁸ For example, we no longer take Newton’s gravitational force as a vera causa—‘true cause’—of the phenomena it was intended to explain (qua action at a distance between point masses). Instead, we understand gravitational force in terms of stress-energy tensor in relativity theory, causing planets to orbit around the Sun by warping spacetime. To echo Cohen and Callender, the natural kind ‘gravitational force’ has changed since Newton, and with it, inevitably, our understanding of the simplicity of Newton’s system and its laws has changed too. Moreover, we know that to explain planetary motion, a lot more assumptions are necessary than just positing Newton’s law of gravity. ⁹ Recently, Eleanor Knox (2014) has argued for the superiority of Newton-Cartan geometrized gravitation over the orthodox Newtonian gravitation as capturing ‘the best spacetime setting for Newtonian gravitation, judged by its own light. In fact, the move to a curved spacetime setting for NG [Newtonian gravitation] is motivated by much the same reasoning that usually impels us to insist that Neo-Newtonian spacetime, and not Newton’s absolute space, is the best setting for Newtonian gravity’ (2014, 864).

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terms of integrating the Lagrangian function of the variables of state over all the possible paths a mass point might take, and choosing the one with the smallest value. So here we have yet another example of how the standard of simplicity has been subject to different interpretations across the history of science. Hamilton’s principle can be extended to quantum mechanics. And when it comes to quantum mechanics itself, and to the so-called fine-structure Hamiltonian defining the state of a quantum mechanical system, mass is no longer a fixed and immutable natural property, because it is subject to the relativistic correction of the rest-mass energy (me c²). Turning to Einstein’s famous equation (E ¼ mc2 ), as Marc Lange has persuasively argued, the equation itself does not tell us the ‘rate of exchange’ between mass and energy. Einstein’s equation in fact relates a body’s rest mass (which is frame-independent, or Lorentz invariant) with the body’s energy (which, on the contrary, depends on speed and hence on the inertial frame; i.e. it is not Lorentz invariant).¹⁰ As these brief historical remarks suggest, it would be odd to conclude that the Lewisian standards of simplicity, strength, and balance are fixed once and for all, and belong to some Limited Oracular Perfect Physicist. Instead, they are standards that have been interpreted differently across the history of science, from Lavoisier’s Best System to Einstein’s Best System, where mass conservation has equally featured as a fundamental law of nature. And the challenge for the Lewisian BSA is then to preserve its account of lawhood despite these perspectival changes in the standards of simplicity, strength, and balance. Momentum conservation too (another jigsaw piece in any Lewisian Best System of nomic facts) appears to have featured across several Best Systems. It still features in contemporary searches at the Large Hadron Collider (LHC), where the large missing ! transverse momentum imbalance pTmiss in multi-jets events might be the signal for possible new Beyond Standard Model particles (e.g. supersymmetric particles, including the neutralino, which is one possible candidate for dark matter). But it would be odd to say that the Best System where the law of momentum conservation features today in Beyond Standard Model searches is one and the same Best System where it originally featured as Descartes’s law of momentum conservation (used to describe elastic collisions between bodies). For once, the strength (information content) of the Lewisian Best System where Descartes’s law of momentum conservation (dead force) featured did not extend to encompass, envisage, or even dream about deep inelastic scattering of fundamental particles and hadronization due to proton–proton collisions at LHC. Perhaps one might argue that the simplicity afforded by the Cartesian Best System (where momentum conservation featured) is not that different from similar simplicity considerations in our current Best System (via, for example, simplified models for particles’ masses and cross-sections).¹¹ ¹⁰ Lange (2001). ¹¹ Simplified models are increasingly used in contemporary high-energy physics as a useful interface between theoretical physicists and experimentalists. They are neither models of the data nor theoretical

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Yet, the Lewisian standard of strength is undoubtedly very different in the two cases, because it reflects the richer informational state of contemporary particle physics. These brief historical remarks should suffice to show that across the history of science there have been more than one Best System where both mass conservation and momentum conservation have featured as ‘laws’ (and, indeed, fundamental ones). And, second, that these different Best Systems resort to different interpretations of the (Lewisian) standards of simplicity and strength. For a glance at the history of the physical sciences soon reveals cracks and discontinuities, shaky foundations, and rival views of simplicity (Newton vs. Cartan vs. Hamilton) and strength (Descartes’s elastic collisions between bodies vs. LHC smashing protons) even in our most promising candidates for the Lewisian title of Best System. What has gone wrong with these examples? Two main lessons emerge, in my view, from these remarks. First, any Best System account of lawhood that intends to explain nomic facts about mass and momentum (across different Best Systems in which they have historically featured) owes us an account of lawhood in terms of evolving standards of simplicity and strength. Second, something ought to be said about the changing standards of simplicity and strength that are meant to define the Best System. David Lewis famously dreaded the suggestion that standards might be dependent on us in any genuine sense. He strived to shelter his account of lawhood from what he described as the lunacy of the ratbag idealist.¹² The lunatic idealist would claim that laws change by changing our ways of thinking about them, or our ways of thinking about what counts as the best system. Lewis acknowledged that in some ways the standards of simplicity and strength are a psychological matter. But he mitigated the possible menace coming from considerations of this kind, with the view that ‘if nature is kind to us,’ the Best System will be robustly so. The robustly Best System will come out far ahead of its rivals, no matter what standards of simplicity, strength, and balance might be in place.¹³ But one does not need to go as far as envisaging a lunatic idealist (after all, there are not many of them around these days). A brief look at the history of science suffices to show how Lewis’s account is vulnerable to the charge of subjectivity. Ned Hall has recently invited BSA supporters to offer a version of reductionism that gets much more serious than Lewis ever did both about what, precisely, the standards are for judging candidate systems, and about why—given the reductionist’s metaphysical commitments—those ought to be the standards . . . In addition, the debate needs a shift in methodological priorities that places much less emphasis on intuitions

models and are simplified in that they focus on only a few simple kinematic parameters such as particles’ masses and their decay products in collisions. ¹² Lewis (1994, 479). I have discussed Lewis’s considered reply to this lunacy in Massimi (2017). ¹³ Lewis (1994, 479).

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about hypothetical cases, and much more emphasis on attending to the distinctive sort of work that a concept of ‘law of nature’ performs in actual scientific practice.¹⁴

In what follows, I take on board Hall’s invitation to change methodological priorities and place more emphasis on the role of laws in actual scientific practice. Hence, I take some preliminary steps towards amending Humeanism about laws so as to reflect the historical evolution that our beloved laws of nature are subject to. I ultimately defend a perspectivalist version of the Best System Account, which builds upon (and hopefully improves on) two somehow similar moves already made in the literature by Cohen/Callender, and Halpin, respectively. My perspectivalist version of BSA is motivated by three main problems affecting Lewis’s view: (1) Subjectivity. Both in the original formulation and in subsequent analyses, Lewis often referred to simplicity and strength as ‘partly a matter of psychology.’¹⁵ An improved version of BSA should avoid subjectivity so understood while at the same time explain Lewis’s original intuition that standards are to some extent dependent on us (although not necessarily on our psychological make-up). (2) Realism. Humeanism about laws implies realism about sparse natural properties composing the Humean mosaic of non-modal facts. But since to be a law is to feature as an axiom or a theorem in the best system; and since as the remarks above suggest, there is more than one Best System historically for the same law, how can we be realist about laws that feature across different Best Systems? (3) Nomic necessity. A pressing complaint against Lewis’s BSA is its inability to deliver on nomic necessity. Can an amended version of BSA improve on the score of nomic necessity? Obviously, by nomic necessity here I mean some suitable notion of nomic necessity, which does not beg the question against the Humeans by appealing to dispositional essential properties or necessitation relations among categorical properties. In Section 2, I present two possible answers to these three problems. These answers have recently been proposed with an eye to improving on Lewis’s BSA. In both cases, Lewis’s ontology of natural properties is retained, but Humean Supervenience rejected. The first answer is in terms of a relativized Best System Account. The second stresses the importance of a perspectivalist Best System Account. I review their common ground and their points of divergence in Section 2. In Section 3, I present my own perspectivalist take on BSA. I clarify how it differs from the two other accounts discussed in Section 2, and most importantly, I illustrate how it can tackle the three aforementioned problems of subjectivity, realism, and nomic necessity.

¹⁴ Hall (2015, 275).

¹⁵ Lewis (1973) and (1994, 479), respectively.

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2. Relativized BSA or Perspectivalist BSA? Two friendly amendments to Lewis’s BSA have been offered in the recent literature. The first, by Cohen and Callender, proposes to relativize BSA. The second, by Halpin, gives BSA a perspectivalist twist. I briefly review both in this section with an eye to assessing their respective success in answering the three outstanding issues of subjectivity, realism, and nomic necessity. The main motivation behind Cohen and Callender’s (2009) relativized BSA is to avoid what they call the problem with inter-system comparison of strength, simplicity, and balance. Namely, to assess the best system among many available ones, we ought to be able to compare their respective standards of simplicity, strength and balance. Yet, BSA involves immanent notions of simplicity, strength, and balance. Thus, systems, whose basic predicates and natural kinds terms ultimately differ, cannot be compared or evaluated on the basis of how they score on simplicity, strength, and balance. An additional motivation for Cohen and Callender’s relativized BSA is to allow for laws in the special sciences (I will not pursue this second motivation here, although it has been widely discussed in the literature).¹⁶ The problem of inter-system comparison does not just concern Goodmanian scenarios where ‘grue’ and ‘bleen’ might happen to be the simple natural predicates. It is instead a compelling problem arising from scientific practice. For a quick glance at the history of science soon reveals a plethora of ways of carving nature at its joints with profound consequences for what counts as a law of nature.¹⁷ Hence, the need for a relativized Best System Account (or relativized Mill-Ramsey-Lewis, MRL as they call it), which makes lawhood ‘relative to a choice among equally available alternatives.’¹⁸ Even if the standards of simplicity, strength, and balance are vague and very sensitive to the particular choice of predicates and kinds at work in any given system, it is still possible to assess, identify, and choose the best system relative to a specific choice of predicates and kinds. This move has significant implications for assessing the classical objection of subjectivity leveled against Lewis. For under Cohen and Callender’s relativized BSA, it is not the content of any law that is at risk of subjectivity; but instead its status as law. For example, Newton’s law of universal gravitation is relative to a given

¹⁶ See Backmann and Reutlinger (2014) and Schrenk (2014). ¹⁷ ‘Our own scientists have axiomatised phenomenological thermodynamics in a variety of ways, some using (e.g.) heat as a category and others not. The laws in each system can differ as a result. Is one axiomatization right and the other wrong because heat is or is not in that distinguished set? . . . Sticking with the thermodynamics example, consider the macrovariable temperature and its changing meanings and extensions. Originally identified with felt hotness, it then was identified with empirical temperature, absolute temperature, and now statistical mechanical temperature. In each case a change in carving accompanied a change in laws. What macrovariables we choose to systematize with appears remarkably pliable. Some, like entropy, the Gibbs free energy, and more, are even somewhat gruesome’ (Cohen and Callender 2009, 17). ¹⁸ Cohen and Callender (2009, 20).

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choice of predicates and kinds in classical mechanics (e.g. thinking of bodies as pointmasses; gravitational force as an action at a distance force; and so on). Yet, the behavior of bodies described by Newton’s law is in no way subjective. Relativized BSA eschews subjectivity also at another level. For laws do not depend for their existence on human subjects. There can be laws (relative to a given choice of kinds K) in worlds without human beings insofar as they feature in the immanently Best System for that world relative to K.¹⁹ In other words, if there were an immanently Best System in world w relative to K (even in the absence of any actual human beings axiomating knowledge according to kinds K), there would be laws in w (relative to K). Relativized BSA acknowledges that there is no Nagelian ‘view from nowhere’ when it comes to lawhood, and it offers ‘a picture of laws that is essentially perspectival’ without the perspectives in question being themselves subject-dependent. This view improves on Lewis’s BSA also on the score of realism. For it chimes with a variety of more modest realist views, ranging from Kitcher’s modest realism, to Dupré’s promiscuous realism, to Putnam’s internal realism. As soon as we recognize that there are many possible ways of carving the world at its joints, relative to the research interests of given communities at given times, our realist commitments too ought to be recalibrated to allow for a pluralism of kinds and ensuing relativity of lawhood. The Humean mosaic of sparse natural properties has to make room for a bewildering variety of kinds. Relativized kinds (rather than God’s-eye-like natural properties) become the minimal ontological unit for our realist commitments, according to the defender of relativized BSA.²⁰ Unsurprisingly maybe, given the Kuhnian flavor of the position, nomic necessity remains an open problem and an ongoing concern for the defender of relativized BSA. So any friendly amendment to Lewis’s BSA ought to improve on relativized BSA on the score of nomic necessity. The aforementioned reference to the perspectival nature of the laws emerging from Cohen and Callender’s account is not casual. For both authors acknowledge their debt to a similar perspectival amendment to Lewis’s view offered by Halpin,²¹ to whom I turn next. Despite significant analogies between Cohen–Callender relativized BSA and Halpin’s perspectivalist BSA, there are also significant differences in the way in which perspective-dependence is understood, which in turn bears on important issues such as nomic necessity, for example. Like Cohen and Callender, John Halpin’s ‘perspectival best system account’ (PBSA) too takes as its starting point the situated and contextual nature of the standards defining the Best System. Knowing the laws of nature is to comprehend a best system ‘with respect to the context of our own conceptual lights.’²² But while

¹⁹ Cohen and Callender (2009, 20). ²⁰ ‘The relativized MRLer, in other words, must be a kind of Carnapian or Kuhnian with respect to theory change, explaining the change from one theory to another as always the result of explanatory/ pragmatic needs and not rational compulsion’ (Cohen and Callender 2009, 29). ²¹ Halpin (2003). ²² Halpin (2003, 142).

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Cohen and Callender understand contextuality in terms of relativizing the Best System Account, Halpin analyses contextuality in perspectivalist terms. PBSA tells us that a best system is very much a matter of perspective. But it is important to get clear about what ‘perspective’ means in Halpin’s use of the word. Perspective is not here understood along the lines of Ron Giere’s ‘scientific perspectives’ as hierarchies of models (from data models to theoretical models), which fit data.²³ Instead, Halpin uses the term ‘perspective’ to denote an idealized continuation of current scientific practice, which would deliver the best system as an aim of science.²⁴ More to the point, to be a law under PBSA is not only to describe the occurrence of regular patterns of natural properties; but it is also a matter of an observer occupying a certain perspective on the Humean mosaic. This feature implies a degree of ‘indeterminateness’ in that laws can only be defined when a ‘complete perspective, i.e. an idealized continuation of actual scientific culture’ is specified.²⁵ Halpin distinguishes among three main perspectives: what he calls the standard perspective, the stipulative perspective, and the philosopher’s perspective. These are different perspectives on our actual world and its Humean mosaic, which result in different best systems of laws. Halpin invites us to think of possible, non-actual other worlds and what the best system of laws would be in these other possible worlds. The standard perspective tells us that the laws in other possible worlds would be nothing other than our own current laws in our actual world. The stipulative perspective tells us that the laws in other possible worlds are stipulated by taking, for example, an ideal approximation to our current world and its laws as the best system. The philosopher’s perspective enjoins us to think of each possible world as having its own best system of laws, without taking our own actual world and projecting its laws into other worlds. Thus, the philosopher’s perspective is said to be the more objective in providing the best system of laws for any possible world. Which perspective to adopt depends to a large degree on the interests of the observer. As Rachel Briggs has observed, if the observer is interested chiefly in building models for the laws at her own world, she can adopt the standard perspective, under which all counterfactual worlds are stipulated to have the laws of her world . . . On the other hand, if the observer is interested chiefly in explaining how the categorical facts determine the laws, she can adopt the philosopher’s perspective, under which each world is assigned the laws that it would have, were it actual.²⁶

Hence, PBSA improves upon Lewis’s BSA on the score of subjectivity. For to be a law is not a matter of being regarded as such, according to subjective or psychological standards. The subjectivity affecting BSA can instead be understood in perspectival ²³ Giere (2006). ²⁴ ‘I should rather see ideal science in this latter way: scientists with time to explore beyond what would normally be practical, would in the long run get to a best system. That would be enough for me. It is enough to say that a best systematization is an aim of science, even if it is not the only aim’ (Halpin 2003, 162). ²⁵ Halpin (2003, 151–2). ²⁶ Briggs (2009, 87).

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terms. It is a product of the observer’s different interests (and different vantage points) when thinking about the best way of systematizing the Humean mosaic: either from some sort of God’s-eye view (the philosopher’s perspective), or from our own current standpoint (the standard and stipulative perspective). This feature of PBSA is important for realism about laws. For PBSA agrees that laws ‘are “there” to be discovered, are independent of any actual theory, though their reality is relative to perspective.’²⁷ In a truly empiricist spirit, PBSA takes nature to be teeming with lawful regularities. These regularities exist and would have existed even if we had not existed. Indeed, ‘worlds without thinking beings have laws. These laws might be projected from our own world (the Standard or Stipulative perspectives) or be the best system of that world (on the Philosopher’s Perspective).’²⁸ However, PBSA departs from Lewis’s ontology of natural properties and natural predicates, precisely because it rejects Lewis’s assumption about one single Best System. By going perspectival, more than one best system is allowed. PBSA defends then a peculiar kind of realism about laws: it is Humean in its fundamental ontology of sparse natural properties; but it rejects Humean supervenience because laws in other worlds w* do not necessarily supervene on the Humean mosaic of actual natural properties at our own world w. More to the point, what laws there are at any other possible world w* is indeterminate until a suitable perspective is specified. Halpin maintains that these perspective-dependent projections of our own laws onto other possible worlds w* improve on Lewis’s BSA also on the score of nomic necessity. Laws under Lewis’s BSA are just factual occurrences of regular events. No necessity is there to be found. But PBSA promises to deliver on nomic necessity: the nomic necessity of the laws is said to be a consequence of perspective-dependent projections.²⁹ As I understand it, nomic necessity under PBSA would then be nothing but the resilience of our own laws projected (under either the Standard or Stipulative perspective) onto other possible worlds w*. These perspective-dependent projections fulfill some practical requirements. When thinking about counterfactual worlds w*, it is just easier and more convenient for us to project antecedent conditions that are akin to the ones in our own current world w, and to expect the consequents to follow according to lawful patterns holding here at world w. Nomic necessity, in Halpin’s account, is downstream of this practicality. It is not the product of any metaphysics of dispositions, or of necessitation relations between categorical properties. In a truly Humean fashion, nomic necessity is nothing but the necessity with which

²⁷ Halpin (2003, 163). ²⁸ Halpin (2003, 164). ²⁹ ‘Laws are necessary in the sense that we tend to project them onto hypothetical situations. At the heart of nomic necessity, then, is the practicality of perspective dependence. (It would be impractical to use laws other than our own to project out possible sets of initial conditions—for example, it would not be particularly useful to project out initial conditions in accordance with principles under which gold is as hard as diamond). As I see it, then, the Humean can explain nomic necessity by reference to this practicality. It is worth stressing this result. It allows the proponent of the PBSA to make sense of modal character without compromising empiricism’ (Halpin 2003, 160).

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past experience brings along with it the expectation that the same lawful patterns occur in other possible worlds (via perspective-dependent projections). In Section 3, I lay out a new version of perspectival Best System Account, which I am going to indicate as npBSA (to distinguish it from Halpin’s PBSA). I highlight similarities and points of departure both with RBSA and PBSA, and explain how npBSA has the potential to address the three problems of subjectivity, realism, and nomic necessity in a novel way.

3. A Novel Perspectival Best System Account of Lawhood What is to be said about the two friendly amendments to Lewis’s Best System Account discussed in the previous section? They both strive to improve on the standard BSA on various counts. And they are both appealing in zooming into scientific practice and bringing to the table Kuhnian considerations from the history of science. Yet, in my view, both fall short of delivering on the full potential of the lessons coming from the history of science and scientific practice. And for different reasons, too. RBSA falls prey of the same diagnosis made for Lewis’s BSA at the end of Section 1. It puts the metaphysical cart in front of the epistemological horse. For Lewis’s emphasis on sparse natural properties is here replaced by relativized kinds, with lawhood defined with respect to them. Thus, in the end it is the relativity of our scientific kinds across the history of science that dictates the need for a relativized BSA. Lawhood is once again at the mercy of ontological considerations. A different way of vindicating what is undoubtedly the important Kuhnian motivation behind RBSA places emphasis not on kinds (or predicates), but on the standards used to assess the best system at any historical time. In other words, a different way of thinking about the Kuhnian move behind RBSA is in terms of the relativity of the standards of simplicity, strength, and balance across time and scientific communities (rather than kinds). Second, something has to be said about nomic necessity and how laws in BSA do not become subjective (the menace of the ratbag idealist, notwithstanding) even if the standards of simplicity, strength, and balance might well change over time and across communities. A revised version of BSA has to make room for the standards to vary over time and yet deliver laws that are resilient to theory change. Halpin’s perspectival account of BSA is a very welcome move in this direction. For it cashes out lawhood in terms of different perspectival standpoints and introduces nomic necessity as a consequence of the practicality of perspective-dependent projections of laws. Yet Halpin’s account too does not give history of science its due, in my view. For ‘perspectives’ are not understood as real scientific perspectives endorsed by real communities across the history of science. They are instead understood as an idealized continuation of actual scientific culture. The problem, as I see it, with

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this quasi-Peircean take on ‘perspective’ is that it rides roughshod over the historical contingencies affecting the Lewisian standards of simplicity, strength, and balance. Historical contingencies are the primary culprit for the contextual and situated nature of the standards employed at any time to define what counts as ‘the best system.’ More to the point, the three perspectival standpoints (standard perspective, stipulative perspective, and the philosopher’s perspective) take either our current perspective (or some God’s-eye view in the case of the philosopher’s perspective) as the privileged standpoint for projecting laws onto other possible worlds. But our view from here now should not beg epistemic priority on other historical views from there then. Given that standards are contextual, a bona fide perspectival approach should deliver a fully dynamic view of how standards have evolved and changed from there then (say, at Lavoisier’s time) to here now (say, contemporary physics on conservation of mass). In what follows, I take some preliminary steps towards cashing out a novel perspectival view on the Best System Account, which takes perspectives as real scientific perspectives of real historical communities across time. My account shares with both RBSA and PBSA the same rationale for amending Lewis’s BSA (namely, attention to scientific practice and perspectival understanding of contextuality). Yet it gives novel answers to the old problems of subjectivity, realism, and nomic necessity. Let us start with three definitions, just to get clear on the main notions I am going to use henceforth. Scientific perspective (sp): A scientific perspective sp is the actual—historically and intellectually situated—scientific practice of a real scientific community at a given historical time. Scientific practice should here be understood to include: (i) the body of scientific knowledge claims advanced by the scientific community at the time; (ii) the experimental, theoretical, and technological resources available to the scientific community at the time to reliably make those scientific knowledge claims; and (iii) second-order (methodological-epistemic) claims that can justify the scientific knowledge claims advanced. Metaphysical, philosophical, religious beliefs that might have been present at the time in the community do not count as part of a scientific perspective. For they cannot explain how the community comes to reliably make or justify those scientific knowledge claims.³⁰

³⁰ I do not have the space to expand on this point but suffice to say that I am here endorsing a twocomponent view of knowledge claims. Reliabilism explains how the community comes to true knowledge claims via certain experimental, theoretical, and technological resources available at the time. Perspectival coherentism (to use Sosa’s expression), in turn, justifies those knowledge claims as part of an epistemic perspective of the community at the time. The epistemic perspective includes not just firstorder knowledge claims about the objects under investigation, but also second-order methodological claims that justify the reliability of the experimental, theoretical, and technological resources used to make the first-order knowledge claims (see Massimi 2012 for more details on this point). The point of this two-component view (following up on Sosa 1991) is to exclude Kuhnian scenarios where, say, Kepler’s neo-Platonism might be regarded as contributing (either to the truth or to the justification) of his knowledge claims about Copernicanism. Kepler’s neo-Platonic beliefs, much as they were present and influential at the time, did not play a direct role in establishing either the truth or the scientific justification for knowledge claims about Copernicanism. Tycho Brahe’s observational data and Kepler’s

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Simplicity, Strength, and Balance at sp (SSBsp): Simplicity, strength, and balance are always contextual standards, typical of a given scientific perspective sp at a given historical time. Different scientific perspectives deploy slightly different sets of standards (SSBsp₁, SSBsp₂, SSBsp₃, . . . —more on this point in the next definition). The perspectivalist of the kind here envisaged understands SSBsp as standards of performance adequacy. They do not define either the truth or the justification for the scientific knowledge claims advanced within a scientific perspective. Instead they monitor the ongoing performance adequacy of the scientific knowledge claims at stake. Namely, these standards are used to monitor how well those scientific knowledge claims serve the epistemic needs of the original community who advanced them, and the needs of any subsequent scientific community that inherits them from the earlier scientific community. Standards of performance adequacy in sp: Simplicity and strength are just two of the standards of performance adequacy employed by scientific communities to monitor how well scientific knowledge claims serve their epistemic needs. The list is much longer and open-ended (e.g. consistency, accuracy, predictive power, explanatory power, just to name a few).³¹ Insofar as the epistemic needs of scientific communities tend to be similar across time and across scientific perspectives, these standards too tend to stay the same across time and across scientific perspectives. Simplicity was a value for Newton no less than for Lavoisier. As such, these standards are constitutive of scientific inquiry across historically and intellectually situated scientific perspectives. This is important because these standards allow scientific communities to assess the ongoing performance of scientific knowledge claims across time. Scientific knowledge claims concerning specific laws of nature that continue to fare well on the scores of simplicity, strength, and their balance over time are justifiably retained in subsequent scientific perspectives; they are

laws (and all the experimental, theoretical, and technological resources to get to them) should instead be counted as part of the epistemic perspective of the community at the time. Thus, my understanding of scientific perspective is closer to Giere’s (2006) than it is to Kuhn’s scientific paradigm (which does include metaphysical and philosophical beliefs among others). ³¹ The attentive reader will not have failed to note the Kuhnian flavor of this list, which is somehow reminiscent of what Kuhn called the ‘values’ part of a disciplinary matrix (see Kuhn 1970, 184–5): ‘Usually they are more widely shared among different communities than either symbolic generalizations or models, and they do much to provide a sense of community to natural scientists as a whole. Though they function at all times, their particular importance emerges when the members of a particular community must identify crisis, or later choose between incompatible ways of practicing their discipline. Probably the most deeply held values concern predictions: they should be accurate; quantitative predictions are preferable to qualitative ones . . . There are also . . . values to be used in judging whole theories . . . they should be simple, self-consistent, and plausible, that is compatible with other theories currently deployed . . . values may be shared by men who differ in their applications . . . judgments of simplicity, consistency, plausibility and so on often vary greatly from individual to individual . . . In short, though values are widely shared by scientists and though commitment to them is both deep and constitutive of science, the application of values is sometimes considerably affected by the features of individual personality and biography that differentiate the members of the group.’ In my view, it is this latter feature of Kuhn’s characterization of values such as simplicity (i.e. their variation in application from individual to individual, their being dependent on individual personality and biography) that Lewis was at pains to avoid (and negatively branded as the lunacy of the ratbag idealist). So, to improve on Lewis’s BSA without falling back into the lunacy he dreaded most, something has to be said about how these values are both constitutive of science, while also changing over time.

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discarded, otherwise.³² Yet there is a sense in which these standards also vary, not among individuals, as Kuhn had it, but among scientific perspectives. Thus, while deeply constitutive of scientific inquiry and fairly stable across theory change, standards of performance adequacy are also subject to interpretive shifts across scientific perspectives. Our perspectival account of BSA ought to reflect this twofold feature.

With these definitions in place, we can now give a first stab at defining a novel perspectival Best System Account of lawhood as follows: (npBSA) Given a scientific perspective sp, and given SSBsp qua standards of performance adequacy, laws of nature are axioms or theorems of the perspectival series of best systems, which satisfy SSBsp₁, SSBsp₂, SSBsp₃, and so forth. (npBSA) improves on Lewis’s BSA on several counts. First, it avoids the implausible ‘view from nowhere’ that affects Lewis’s definition of best system, with all ensuing objections that have been leveled against it (most famously, van Fraassen’s objection against Lewis’s ‘eschatology of science’). There is not just a single Best System; a privileged philosopher’s Best System à la Halpin (or what Ned Hall calls Lewis’s LOPP, Limited Oracular Perfect Physicist). Instead, there are several BSAs across the history of science, as many as there are clearly identifiable scientific perspectives reinterpreting the standards of simplicity, strength, and their balance. Second, (npBSA) takes on board the Lewisian vagueness about simplicity and strength and interprets it not as a matter of psychology, but as perspectival standards of performance adequacy endorsed by real scientific communities. The natural advantage of this move is that it undercuts the possible menace of the ratbag idealist dreaded by Lewis. Moreover, it eschews the problem of subjectivity that haunts Lewis’s BSA. For simplicity, strength, and balance are not to be understood as figments of one’s own psychological make-up. Nor are they by-products of mob psychology either. Given what a scientific perspective is (in terms of making reliable scientific knowledge claims and being able to justify them), and given the role of standards of performance adequacy (in terms of assessing the ongoing performance of scientific knowledge claims vis-à-vis the epistemic needs of a scientific community), no dreaded idealist lunacy looms on the horizon. Third, npBSA stresses (even more than Lewis ever did) the contingency of lawhood on the perspectival series of best systems. And contingency is here understood in terms of contextual or perspective-sensitive standards for the best system at any one time (SSBsp₁, SSBsp₂, SSBsp₃, . . . ). According to npBSA, Newton’s law of gravity is a theorem in the best system at Newton’s time (defined with respect to SSBsp₁, say, with simplicity understood as

³² In this sense, I agree with Kuhn that these values are important to assess eventual crises and to choose between incompatible ways of practicing a discipline. They provide a canon for assessing how well scientific knowledge claims continue to perform over time, or whether they fail by their own lights.

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‘true and sufficient causes’ for phenomena). And it continues to be a theorem in our best system now (say, SSBsp₂, with simplicity no longer interpreted along Newton’s original lines). Mass conservation was an axiom in the best system at Lavoisier’s time (where strength was a large-meshed net that allowed for caloric to feature among non-nomic facts). And it continues to be an axiom in our best system now, despite our standards of simplicity and strength having been drastically recalibrated to exclude the deduction of caloric-related phenomena. Momentum conservation was a law in the Cartesian best system, and it continues to be a law in our current best system, where strength has been redefined to encompass not just elastic collisions in classical mechanics, but also deep inelastic scattering, jet fragmentation, and hadronization in high-energy physics. More importantly, npBSA has also the potential of addressing the problems of realism and nomic necessity. Laws of nature are real because there are genuine regularities in nature, whose behavior can be discovered and expressed in mathematical language (be it differential equations or else) within a system of scientific knowledge claims that satisfies perspectival standards of performance adequacy such as simplicity, strength, and their balance. The perspectival nature of the best system at any given historical time does not affect the reality of the regularities captured by the laws. What Lavoisier observed about weight being conserved in combustion and calcination processes was real then as it is now. What Descartes observed about the collisions of rigid bodies was real then as it still is nowadays when we consider how quarks and gluons get knocked out of protons to form new hadrons. By making simplicity, strength, and balance perspectival standards of performance adequacy, the reality of co-occurrent regularities remains unscathed (no matter in which lawful dress we may deck them out across the history of science). This is an important feature of the present proposal that feeds into a larger project (namely, cashing out perspectival realism in science), whose discussion will have to wait for another occasion. For I want to conclude this brief exposition of npBSA with a discussion of what I consider the main advantage of the position: i.e. the way it retrieves some suitable and non-question-begging form of nomic necessity. As discussed above, the lack of nomic necessity has been a repeated source of complaint against Lewis’s BSA. Halpin’s PBSA set out to improve on Lewis’s account by thinking of nomic necessity in terms of perspective-dependence projections of our own laws into other possible worlds. Yet, in my view, it remains unclear how projecting our laws onto other possible worlds is ever going to bestow modality to those laws. Unless the projectivist account has a pretty convincing story to tell about what makes our own laws modally robust in the first instance, nomic necessity does not accrue for free via perspective-dependence projections. Thus, while I share Halpin’s quasiHumean intuition that the practicality of perspective-dependence is at the heart of nomic necessity, I part my way from Halpin’s understanding of it. For I read ‘practicality of perspective-dependence’ in terms of resilience across perspectival changes in simplicity, strength, and balance (SSBsp) in different best

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systems. In other words, it is the resilience of our laws (such as conservation of mass, momentum conservation, or Newton’s law of gravity) despite the perspectival nature of our standards of simplicity and strength that is testimony to the nomic necessity of these laws. Pragmatic considerations about how these laws have served our predecessors—and continue to serve us so well today—provide good reasons for taking the underlying empirical regularities as nomologically necessary (perspectival simplicity, strength, and balance notwithstanding). In other words, no matter how the standards of simplicity and strength (individually and/or jointly) change across scientific perspectives, it is ultimately the (always renegotiable) balance between these two revisable standards that allows us to reliably trace nomological continuity across historical periods and scientific perspectives. Humeans are usually faced with a dilemma. Either (1) a causal-glue-free mosaic of natural properties, which can only make factual (but not modal) claims about what has been, is, and will be the case (e.g. ‘there has not been, is not, and will not be a perpetual motion machine,’ as opposed to ‘there could not be a perpetual motion machine’). Or, (2) capitulate that laws do not supervene on matters of fact, and worlds identical as to matters of fact can yet be different in their laws depending on whether property F-ness necessitates G-ness in world w₁ but not in w₁, as Necessitarians would argue.³³ In reply, I suggest that Humeans can avail themselves of an alternative way of thinking about nomic necessity, whereby nomic necessity is neither an emanative effect of dispositional essentialism, nor a metaphysical liability, as with Armstrong’s Necessitarian account of laws. For nomic necessity might well be nothing over and above the resilience of laws across different scientific perspectives. There is no need for either postulating inexplicable necessitarian relations between universal properties; or a causal glue between, say, mass m and velocity v (when it comes to the law of momentum conservation). Laws of nature are nomologically necessary as long as they feature as axioms or theorems in a perspectival series of best systems across different scientific communities and times. It is the (always renegotiable) balance between the revisable perspectival standards of simplicity and strength from sp₁ to sp₂ and so forth, which ultimately secures the nomological necessity of our fundamental laws across scientific perspectives. Mass conservation or momentum conservation are necessary because they resiliently feature in our perspectival series of Best Systems, no matter how our epistemic needs might have changed across centuries, and our perspectival standards with them. Of course, this nomic necessity is contingent on us (as Lewis would have it), and on our historical, perspectival Best Systems (as I would add). Critics are going to remain unimpressed by this move. For nomic necessity is usually associated with either some metaphysical machinery designed to deliver on it; or with some

³³ For this anti-Humean argument, see Carroll (1990).

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invariance under counterfactual antecedents. Weak as my perspectival proposal might sound, it strikes me as a historically more realistic option than nomic necessity being contingent on which categorical property necessitates which other in an Armstrongian world or which counterfactuals remain invariant across possible worlds.³⁴

³⁴ I thank the editors for the opportunity to contribute to this volume and for careful and constructive comments. I am very grateful to Marc Lange for reading an earlier draft and providing detailed and stimulating comments. This article is part of a project has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement European Consolidator Grant H2020-ERC-2014-CoG 647272 Perspectival Realism. Science, Knowledge, and Truth from a Human Vantage Point).

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9 Laws An Invariance-Based Account James Woodward

1. Introduction This chapter defends an account that links laws of nature to invariance (aka stability, robustness). Laws are generalizations¹ about repeatable relationships² that are invariant over variations in initial and other sorts of conditions, at least within an appropriate range of such variations—invariant in the sense that laws will or would continue to hold under such variations. Alternatively, laws are generalizations that exhibit a certain sort of independence from initial conditions, where initial conditions themselves are understood as meeting, at least ideally, certain sorts of (additional) independence requirements. I have defended an invariance-based account of laws elsewhere³ and broadly similar accounts have been proposed by other writers,⁴ although their treatments differ in detail from mine. As far as I can tell, such accounts have been less influential than several alternatives, including Best Systems analyses⁵ and accounts that either attempt to ‘ground’ laws in metaphysically special entities⁶ (e.g., relations between

¹ The word ‘law’ is used in both philosophy and science to refer both to (i) generalizations, formulated mathematically or in some other representational format, as when one says that Maxwell’s equations are laws of nature, and (ii) to whatever it is in nature that ‘corresponds’ to these generalizations. For the most part, I adopt the former usage on the grounds that it is less likely to beg important questions about what are the ‘objective correlates’ of laws in sense (i) above. ² Laws are often taken (not just by philosophers but by scientists—see the quotations from Wigner below) to describe ‘regularities.’ My talk of repeatable relationships is meant to avoid some of the problematic features of this view but little will turn on this in what follows. ³ Woodward (2003, 2013). ⁴ Mitchell (2000) and Lange (2009b). ⁵ Lewis (1986b). ⁶ Sometimes called ‘non-Humean’ entities or properties or ‘whatnots’ (Lewis 1986b). Although I will sometimes use similar terminology, it is not entirely perspicuous. In my view, there is nothing problematic about such ordinary language claims as ‘salt has a disposition to dissolve in water’—problems only arise when this disposition is invested with various other features and accorded the explanatory role of ‘grounding’ laws. Insofar as there is an objection to ‘non-Humean stuff,’ it ought to be directed at this latter use of disposition talk.

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universals, dispositions possessed essentially) or to replace laws with claims about such entities.⁷ I thus begin with some general remarks about why invariance-based accounts of laws deserve more attention.

1.1 They fit scientific practice well (or at least better than alternatives). Explicit discussion of laws, at least in physics, commonly links these to symmetry and invariance principles, as illustrated by the remarks of Eugene Wigner discussed in more detail in Section 2. It is natural to think of symmetry principles as special cases of a more general invariance-based conception of laws—see below. By contrast, in my view and as argued by others,⁸ the idea that laws must satisfy symmetry/invariance conditions does not emerge very naturally, if at all, from alternative accounts of laws—instead it has to be imposed, to the extent that it can, by hand.

1.2 Laws are often described by having a kind of necessity or inviolability. Neither we nor the rest of nature can ‘break’ or avoid them, at least in regimes in which they hold. However, this notion of necessity, if intelligible at all, stands in need of explication, and alternatives to invariance- based accounts have not been very successful at this. Some⁹ claim that the necessity of laws is a kind of metaphysical necessity, but, in addition to its obscurity, current statements of this idea do not fit well with how scientists reason about laws. Scientists seem quite willing to consider scenarios in which such alleged metaphysical necessities are violated—e.g., what the trajectories of the planets would be if gravity conformed to an inverse cube law.¹⁰ ‘Metaphysical necessity’ seems too strong a notion and too far removed from what might be established by empirical investigation to capture the kind of necessity possessed by laws. Best Systems analyses have the opposite problem: they take laws to be regularities described by axioms or theorems in a deductive systemization involving a best balance of simplicity and strength imposed on a Humean basis specified in terms of spatiotemporal relations and perfectly natural monadic properties, characterized nonmodally. Although defenders can stipulate that what it is for a generalization to be ‘physically necessary’ is just for it to be such an axiom or theorem, it is not obvious what this status has to do with notions of inviolability or necessity or, more generally, why one should expect the axioms and theorems of the Best System Account (BSA) to be invariant in the sense described above. On the contrary, the BSA threatens to make laws too sensitive to (non-invariant with respect to) initial condition information.¹¹ ⁷ Mumford (2004a). ⁸ E.g. Lange (2004). ⁹ E.g. Bird (2007). ¹⁰ This point is emphasized by Lange (2009b). ¹¹ See Woodward (2003) and below for additional discussion. On the invariance-based account, laws are those generalizations that continue to hold under a range of different initial conditions. But sufficiently

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By contrast, invariance-based accounts provide a naturalistic, scientifically respectable and non-mysterious treatment of what non-violability and physical necessity amount to: this just amounts to the claim that within the domain of invariance of a law there are no initial and background conditions that might be realized—nothing that might be done by nature or an experimenter—under which the law will fail to hold.¹² Inviolability, non-breakability, and so on are thus just other names for invariance. Moreover, the invariance of a generalization is something for which we can get empirical evidence.¹³

1.3 Invariance-based accounts combine some of the most plausible elements in both BSA-type accounts, and more ‘metaphysical’ accounts, while rejecting less attractive elements of both accounts. Invariance-based accounts agree with the BSA and disagree with the metaphysical accounts in holding that the postulation of special metaphysical entities is unhelpful. However, like metaphysical accounts, and unlike the BSA, invariance-based accounts do not aspire to provide a reduction of the modal features of laws to non-modal features. Although the invariance-based account can be thought of, in the respects described above, as situated midway between the BSA and metaphysical accounts, it departs from both in a fundamental resect. Both the latter aspire to provide ‘truthmakers’ or ‘metaphysical foundations’ for laws, with this presumably consisting of something like the full Humean basis in the case of the BSA, and the metaphysically special entities and relations in the case of non-Humean theories. As commentators have noted (and, often, complained)¹⁴ the invariance-based account provides no such metaphysical story about truth makers or foundations, either in the form of special entities or a reduction. For many philosophers, the absence of such a story in an account of laws is rather like Hamlet without the prince—the most important element has been left out. Bird, for example, writes that although the invariancebased account makes an ‘important contribution to understanding the superstructure [of how we reason about laws] it does not tell us (nor is it intended to tell us) about the metaphysical underpinnings.’¹⁵ He also suggests that any one of the

different initial conditions will generate significantly different Humean bases, which may require different axioms for their systemization. Thus which generalizations are BSA laws will depend on the actual distribution of initial conditions. This reflects the fact the BSA does not cleanly separate lawful and initial condition information in the way the invariance-based account does. ¹² As explained in more detail below, this is not just the triviality that laws hold when they hold; it rather involves the claim that laws would continue to hold if non-actual conditions (and non-actual processes leading to these conditions) within their domain of invariance were to be realized, where this domain can be given an independent specification—in physics this is often done by specifying length or energy scales in which the law holds. ¹³ Woodward (2013) and Section 4 in this chapter. ¹⁴ See, e.g., Bird (2007) and Psillos (2004). ¹⁵ Bird (2007, 5).

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alternative accounts mentioned above might provide such underpinnings for talk of invariant relationships. In response, I would emphasize that, first, there is a great deal of value in providing a descriptively accurate characterization of the ‘superstructure’ of our thinking about laws and the role they play in scientific theorizing—this superstructure is far more complex and subtle than many philosophers suppose. Second, contrary to what Bird suggests, it is not true that all of various accounts of the metaphysical underpinnings of laws on offer can be made to fit with the invariance-based account. Instead, the invariance-based account captures features of scientific practice not readily captured by any of these alternative accounts. A third and more fundamental consideration is whether, as Bird implies, the invariance-based account not only leaves something out (a story about metaphysical foundations) but also leaves something out that can and needs to be provided. Disciplines other than philosophy (including in particular mathematics and the natural sciences) recognize the possibility of ill-posed problems—problems which as currently formulated rest on mistaken presuppositions or which assume constraints on what counts as a solution that are unsatisfiable or ill-motivated. Sometimes solving a problem requires rejecting one or more of these presuppositions/constraints and/or reconfiguring the problem itself or rethinking what is required to solve it. I’m inclined to think that the problem of identifying truth makers for laws in the sense of ‘truth maker’ that most philosophers have in mind is just such an ill-posed problem. A detailed argument for this claim is beyond the scope of this chapter but I try to briefly suggest (Section 9) why I think it is true.

2. Wigner on Invariance and Independence With this as motivation, I turn to some remarks about laws, invariance, and initial conditions taken from several of the essays in Eugene Wigner.¹⁶ I quote extensively because I want to make detailed use of Wigner’s ideas in elucidating the connection between lawfulness and invariance. Wigner’s point of departure is the contrast between laws and initial conditions. He writes: The world is very complicated and it is clearly impossible for the human mind to understand it completely. Man has therefore devised an artifice which permits the complicated nature of the world to be blamed on something which is called accidental and thus permits him to abstract a domain in which simple laws can be found. The complications are called initial conditions; the domain of regularities, laws of nature. Un-natural as such a division of the world’s structure may appear from a very detached point of view, and probable though it is that the possibility of such a division has its own limits, the underlying abstraction is probably one of the most fruitful ones the human mind has made. It has made the natural sciences possible. ¹⁶ Wigner (1979).

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He also stresses the role of invariance principles in the identification of laws: However, the possibility of isolating the relevant initial conditions would not in itself make possible the discovery of laws of nature. It is, rather, also essential that given the same essential initial conditions, the result will be the same no matter where and when we realize these. This principle can formulated, in the language of initial conditions, as the statement that the absolute position and the absolute time are never essential initial conditions is the first and perhaps the most important theorem of invariance in physics. If it were not for it, it might have been impossible for us to discover laws of nature.

Commenting further on the contrast between laws and initial conditions, Wigner introduces several additional ideas: The fact that initial conditions and laws of nature completely determine the behavior is . . . true in any causal theory. The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter. (My italics)

Wigner also suggests that if there were regularities or relations or constraints among initial conditions, this would suggest that our specification of the laws of nature was inadequate: how can we ascertain that we know all the laws of nature relevant to a set of phenomena? If we do not, we would determine unnecessarily many initial conditions in order to specify the behavior of the object. One way to ascertain this would be to prove that all the initial conditions can be chosen arbitrarily.¹⁷ The minimal set of initial conditions not only does not permit any exact relation between its elements: on the contrary, there is reason to contend that these are, or at some time have been, as random as the externally imposed, gross constraints allow.¹⁸

He explains what he has in mind by reference to Kepler’s laws and Bode’s law. These involve regularities in what we think of as initial conditions. Wigner comments: ‘However, the existence of the regularities in the initial conditions is considered so unsatisfactory that it is felt necessary to show that the regularities are but a consequence of a situation in which there were no regularities.’ As an illustration, he cites a hypothesis advanced by von Weizsacker according to which the ‘solar system consisted of a central star, with a gas in rotation but otherwise in random motion around it.’ Von Weizsacker attempts to deduce regularities like Bode’s law from this hypothesis; thus, in Wigner’s words, ‘Attempt[ing] to show that the apparently organized nature of these initial conditions was preceded by a state in which the uncontrolled initial conditions were random.’ He adds, ‘These are, on the whole, exceptional situations. In most cases, there is no reason to question the random nature of the noncontrolled, or nonspecified, initial conditions.’¹⁹

¹⁷ Wigner (1979, 40).

¹⁸ Wigner (1979, 41).

¹⁹ Wigner (1979, 41).

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From these remarks, we may extract the following picture. In many although perhaps not all cases, science proceeds by (somehow) making a ‘cut’ or ‘separation’ between initial conditions and laws. Wigner does not describe procedures for doing this (nor will I), but he does tell us something about the features that each component (initial conditions versus laws) will possess once we have made the separation—thus describing in part what success in making the separation will consist in. First, the laws are associated with ‘regularities’ that possess invariance properties. Part of what this means is that the laws possess certain symmetries in the sense of returning exactly the same solutions or predictions under certain kinds of transformations in the initial conditions—for example, the laws may tell us that otherwise similar systems which are spatially translated with respect to each other will behave in the same way and similarly for other sorts of transformations. According to Wigner, some minimal degree of invariance of this sort is probably necessary for doing science at all. Laws also possess additional invariance properties which we can think of as generalizing these symmetry-linked invariance conditions: the laws themselves (and the relationships described by the laws) should be stable or continue to hold as the initial conditions vary, at least for some range of such variations, even though the laws will yield different predictions as we combine them with different initial conditions. While the invariance/symmetry of, say, Newton’s gravitational inverse square law under spatial translations implies that two masses separated by a fixed distance will exert the same force on each other if this system undergoes a spatial translation, this more general notion of invariance requires that the law itself—F ¼ Gm1 m2 =r 2 and the relationship it describes—should continue to hold as the magnitudes of the masses and the distance between them vary, at least for some ‘appropriate’ range of such variation (see below for what this means). In other words, we should make the cut between laws and initial conditions in such a way that the laws are such that when conjoined with a range of different initial conditions, they continue to correctly describe the behavior of the systems to which they apply. Moreover, we should formulate individual laws in such a way that they are invariant not just over initial conditions in the sense of values taken by variables explicitly figuring in those laws (m and r in the case of the inverse square law) but under changes in other variables as well—e.g., changes in the colors or shapes of the masses in particular gravitating systems or other sorts of ‘background’ changes. In what follows I will use the notion of invariance under changes in ‘initial conditions’ to include such additional changes as well. To pick up on other aspects of Wigner’s remarks and to make a connection with what will come later, one way of thinking about these invariance conditions is that they require a kind of independence of laws from initial conditions—the cut between laws and initial conditions should be made in such a way that they are ‘independent’ of each other or as ‘separate’ as possible. Of course, in this sort of context,

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‘independence’ does not mean statistical or probabilistic independence,²⁰ at least in any literal sense: laws do not correspond to random variables characterized by joint probability distributions also involving initial conditions. Nor does it seem right to think of the independence in question as a kind of causal independence of any straightforward sort. One possibility suggested both by Wigner’s discussion and some other ideas derived from machine learning described briefly below (Section 6) is to think of the independence in question as a kind of informational independence: we want the split between laws and initial conditions to be chosen in such a way that these are as informationally independent as possible, in the sense that specific information about which initial conditions obtain for a system should not provide information (or at least specific, non-generic information) about the laws that describe that system and conversely. In Wigner’s language there should be ‘no relation’ (at least of a sort we are able to describe) between whatever is specifically true of the initial conditions and whatever is true of the laws and conversely. Of course this requirement needs to be understood in such a way that it is consistent with the same variables characterizing both the laws and initial conditions—otherwise we would not be able to apply the laws to the initial conditions. So in this respect there will be some connection between the laws and initial conditions. Rather the idea is that specific values taken by the initial conditions or specific parameterizations of those conditions or specific relationships characterizing patterns in those conditions should not occur in the laws, if this can be avoided (again, see Section 6). Speaking metaphorically, given the laws, we or nature should be able to independently and freely choose whatever initial conditions we wish to combine them with.²¹ So far we have been talking about the ‘independence’ of laws from initial conditions. Recall, however, that Wigner also describes what are in effect independence conditions applying to the initial conditions themselves—roughly, these ‘should be as random as the externally imposed gross constraints will allow’ with the existence of regularities in initial conditions being considered ‘unsatisfactory.’²² Wigner does not further explain what he means by ‘random’ or by ‘the existence of a regularity in the initial conditions’ but here statistical or quasi-statistical interpretations seem more appropriate: absent specific information to the contrary, there is a default in favor of assuming that initial conditions characterizing different systems or characterizing appropriately distinct elements of the same system (where these are taken at the ‘same time’—that is, on a surface of simultaneity—see below) should be ²⁰ Or at least there is no obvious warrant for such an interpretation. A Bayesian treatment in which it is assumed that there is a joint probability distribution over laws and initial conditions may be possible but I ignore this in what follows. ²¹ In certain speculative multiverse cosmological models, this picture of different sets of laws and different sets of initial conditions being generated independently of one another in different ‘worlds’ may be understood non-metaphorically. ²² Wigner (1979, 41).

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statistically independent of one another. Or, more weakly, one might take the idea to be that such initial conditions should at least be capable of varying independently of each other—there should be no lawful constraints among the initial conditions themselves. Moreover, one might expect that under the right circumstances this possibility of free variation should manifest itself in statistical independence or something close to it, as well as in the ability of experimenters to set up experiments at separated locations realizing ‘arbitrarily’ different initial conditions.²³ If, contrary to this default, a regularity or non-independence is present among the initial conditions, one would expect that there is some further explanation of this, which traces the dependency or regularity back to earlier conditions satisfying the randomness requirement, as illustrated by the hypothesis about the formation of the solar system. This suggestion about initial conditions is allied to assumptions that are sometimes described under such rubrics as the independence of incoming influences or the principle of the common cause, where the latter says that if X does not cause Y and Y does not cause X and they have no common cause Z, one should assume that X and Y are statistically independent. This provides a connection between this portion of Wigner’s remarks and a commonly accepted, if sometimes controversial, methodological maxim. Finally, to avoid misunderstanding, let me emphasize that whether some proposed set of laws (and accompanying sets of claims about initial conditions) actually have the features of invariance and independence described above is always an empirical matter. It is nature and not any kind of a priori or purely formal analysis that settles whether or not some proposed law is or is not invariant under some specified set of changes in initial conditions.

3. Wigner Explicated Further Let me next underscore some additional features of these ‘Wignerian’ ideas. Note first that both initial conditions and laws are characterized via a set of interrelated constraints that must be satisfied together: we choose what to count as a law and what to regard as initial conditions in such a way that these constraints are simultaneously satisfied. In particular, the invariance possessed by laws is not understood as a matter of the law continuing to hold under just any arbitrary counterfactual stipulation but rather as invariance under a much more specific set of changes— changes in initial conditions, where these meet further requirements having to do with independent realizability. Note that the ‘independence’ and the ‘realizability’ component of this requirement themselves have a modal character.²⁴ ²³ The notion of ‘free assignability’ of initial conditions is sometimes used to express both this idea about the absence of relations among initial conditions and their independence from the laws. ²⁴ ‘Realizability’ is a subtle notion that deserves more explication than I can give it here. At least in many cases, it means that the occurrence of the initial condition taken in itself is physically possible. However, in a

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This is one of several respects in which, in the account described above, invariance is understood in a non-reductive way—the (modally committed) notion of a change in an independently realizable initial condition enters into the characterization of invariance. This picture contrasts with one in which one somehow begins with access to information that is completely non-modal in character, and that can be characterized independently of ‘laws,’ identifies regularities from this information, and then promotes some of these to the status of laws on the basis of additional criteria such as those specified by the BSA. Another feature of this invariance-based picture is that the initial conditions are regarded as subject to very different constraints than the laws—roughly, one tries to put as much structure or regularity as possible into the laws (in this sense one looks for ‘simple’ laws), while choosing what counts as initial conditions in such a way that these are as unstructured as possible (if one thinks of this also as simplicity constraint, it has different consequences than the simplicity constraint imposed on laws). One can think of this as a matter of choosing so as to simultaneously satisfy different desiderata (order for laws, disorder for initial conditions) but this choice appears to have a different structure than the ‘trade-off ’ described in the BSA. Also worth noting is Wigner’s pragmatic attitude toward the ‘split’ between laws and initial conditions. Although he describes it as ‘one of the most fruitful divisions that the human mind has made,’ he also says it is an ‘artifice’ and perhaps ‘unnatural’ from a ‘very detached perspective,’ adding ‘that the possibility of such a division has its own limits.’ In a footnote²⁵ he suggests that ‘the artificial nature of the division of information into ‘initial conditions’ and ‘laws of nature’ is perhaps most evident in the realm of cosmology,’ adding that, ‘It is in fact impossible to adduce reasons against the assumption that the laws of nature would be different even in small domains if the universe had a radically different structure.’ In remarking that there may be physical situations or contexts in which the law/ initial condition distinction breaks down or fails to apply, I take Wigner to be suggesting that our ability to make this distinction requires that nature behaves in a way that ‘supports’ the distinction. The distinction thus tracks ‘objective’ features in nature, but these features are not guaranteed to be present in all situations. At the same time, however, Wigner also emphasizes that there is an element of ‘artifice’ in the construction of theories embodying this distinction, with such theories being adopted because they are ‘fruitful’ and not just because nature allows us to construct them. At the risk of putting words in Wigner’s mouth, this suggests that there is a ‘functional’ story²⁶ to be told about role of law/initial condition distinction in our thinking—functional in the sense that this structure or architecture number of cases, no such requirement of physical possibility is imposed on whatever process is modeled as leading to the initial condition in question—e.g., one may model the initial condition as imposed by a Dirac-delta function-like impulse. ²⁵ Wigner (1979, 3). ²⁶ For more on what is meant by a ‘functional’ story in the context of understanding causation, see Woodward (2014).

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is adopted because it is more fruitful in contributing to certain goals than alternative systems of representation that do not involve a separation of laws and initial conditions. (For more on what these goals might be, see Section 5.) If one were to focus entirely on this second feature, having to do with the goal of finding theories with a certain structure, to the exclusion of the first, having to do with the features of nature that permit the construction of theories with this structure, one might be led to a view according to which the law/initial condition contrast is merely something that we ‘make up’ and ‘project’ onto nature. The fact that nature has to behave in certain ways for the attempt to describe it in accord with the law/initial condition contrast to be successful suggests instead that more is going on than mere projection—it points us in the direction of (or suggests that there must be something correct about) a more ‘realistic’ picture of the status of laws. But I suggest that what is supported is (at best) a kind of tempered and restricted realism: we should think of the law/initial condition distinction as a distinction that works well for us, at least often and around here, and we should not neglect the fact that the distinction is employed because there are things we want to do with it—e.g., predict and explain. Like ‘cause,’²⁷ ‘law’ thus has a dual or Janus-faced character; it functions to organize our thinking in a certain way (it has a ‘design’ component) and, when applicable, tracks features of the world. To understand the notion of law one needs to keep both of these aspects in mind. To bring out the significance of this last point, observe that it is certainly possible to describe nature in ways that do not employ a distinction between independent laws and initial conditions. Following Wigner, I will take it that the (or a) major reason for not doing this is that the resulting descriptions are less fruitful or effective than those incorporating the law/initial condition split. Although I lack space to argue for this claim in detail, I take this (the possibility and fruitfulness of the distinction) to be a discovery that occurs, perhaps gradually, in the early modern period, with a key element involving rejection of pictures of nature in which, from a modern perspective, law and initial condition information are intermingled or conflated rather than distinguished and regarded as independent. Thinking, as scholastics and Aristotelians were wont to do, of nature in terms of active powers and tendencies inherent in bodies that lead them to behave as they do and focusing on what happens most frequently or ‘for the most part’ involves just such an intermingling, which needed to be rejected for modern science to take the form that it did.²⁸ A key element in this change was the recognition that the world of common sense and ordinary observation is disorderly in many respects and that this disorder can be regarded as the upshot of often hidden orderly constraints (the laws), interacting

²⁷ See Woodward (2014). ²⁸ What happens most frequently reflects a kind of mixture of which initial conditions are realized in nature most often and the laws applying to these conditions. Thinking in terms of powers which are characterized in terms of their effects also seems to lead in many cases to a mixing or lawful and initial condition information, since the effects depend on which initial conditions are present.

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with disorderly initial conditions, with the former capable of being formulated independently of the latter, so that the disorder can, as it were, be segregated off into the initial conditions. Thinking of laws as in some way ‘external’ to (or independent of) the particular objects they describe, however puzzling this may seem metaphysically, seems to have been a key part of this change in thinking. Let me conclude this section by acknowledging what will be obvious to cognoscenti: the picture I have presented, when construed as a general account of laws in science (or even in physics) is (to put it charitably) hugely oversimplified. To begin with, although Wigner does not explicitly say this, his formulation seems to require (or at least to apply most naturally) to systems in which there is a well-posed ‘initial value’ problem: a so-called Cauchy surface which allows for a complete specification at a time slice of the initial conditions governing the system. (It is these conditions that are supposed to be capable of freely varying or independent of one another. Obviously initial conditions at different time slices will not be independent.) Wigner thinks of the fundamental laws (as many philosophers of science do) as dynamical laws expressed by hyperbolic differential equations specifying the evolution of these systems over time. At present by no means all candidate laws take this form. Some candidate laws are not of evolutionary type and are not expressed by means of hyperbolic differential equations. Relatedly, in focusing, as Wigner does, on invariance under initial conditions, I have completely ignored the role played by boundary conditions, needed for the characterization of many systems and for the solution of the differential equations (the laws) applied to those systems. As emphasized by Wilson, the equations governing the interior of a system may accept only certain boundary conditions and not others;²⁹ indeed, certain boundary conditions (and the physics they implicitly assume) may be inconsistent with the equations governing the interior. The boundary conditions themselves may embody additional modal information (e.g. information about what can be changed independently along the boundary and what cannot), so that again we have a situation in which not all such information is carried by the laws alone. Obviously the simple characterization of invariance-related notions attributed above to Wigner requires elaboration and qualification if it is to be extended to these more complicated situations. I will not try to do this here. One has to begin somewhere.

4. More on Invariance Let me turn now to a closer look at the notion of invariance. One very natural construal is modal or counterfactual: ²⁹ Wilson (1990).

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(9.1) generalization G is invariant if it would continue to hold were certain changes in initial conditions to hold. (This is not the only possible construal. Section 6 will briefly consider an interpretation in terms of informational independence.) An obvious question is then how (9.1) should be understood. If we think in terms of the standard Lewisian possible worlds analysis, we seem to be led to construals like the following: (9.2) In the closest possible worlds in which alternative initial conditions Vi occur, G continues to hold. We then encounter the problem that the Lewisian similarity measure for closeness among possible worlds makes reference to the laws of nature, which include G. We thus seem to have a vicious circle: to evaluate (9.2) it looks as though we need to already know what the laws of the actual world are (and perhaps whether and to what extent such laws are ‘preserved’ under various counterfactual suppositions). How then can we use a notion of invariance understood in terms of such counterfactuals to get a purchase on laws? Thoughts along these lines are presumably part of what underlies the complaint that appealing to invariance to explicate laws of nature is (viciously) ‘circular.’ For example, Psillos argues that it is the laws that determine which variations in initial conditions are physically possible and hence which variations are appropriate for purposes of assessing invariance,³⁰ an assessment which is also endorsed by Bird.³¹ Suppose, to use Psillos’ example, we want to assess the invariance of generalization G. According to Psillos, it would not be appropriate to ask whether G is invariant under some variation that involves a particle moving faster than light since this is a physically impossible variation. But (Psillos then argues) this judgment requires that we have already identified what the laws are, thus landing us in a ‘circle.’ More generally, many philosophers hold that laws are among the ‘truth-makers’ for counterfactuals, so that to the extent invariance is bound up with counterfactuals, we can’t appeal to invariance considerations to elucidate the notion of law. These complaints move much too quickly from the non-reductive character of an invariance-based account to the conclusion that it is epistemically or otherwise circular in a vicious way. First, let me repeat that what invariance requires is stability of a generalization under variation in (independently realizable) initial conditions and not under all possible counterfactual suppositions that philosophers may be willing to make. Thus for the purposes of assessing invariance we need not worry about the truth value of such counterfactuals as ‘if the wires on this table (which are copper) had been insulators . . . then “copper conducts electricity” would not be a “law.”’³²

³⁰ Psillos (2004).

³¹ Bird (2007).

³² Cf. Lange (2009b, 38–9).

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The antecedents of such counterfactuals do not correspond to assumptions about initial conditions, let alone independently realizable ones.³³ Second, our epistemic access to which initial conditions are physically possible, singly or in combination, obviously does not require that we already know what the laws of nature are. One can learn about what is physically possible both from observation of the variation in states and initial conditions that occur naturally and by active experimental manipulation. For example, one can determine empirically that there is no way of realizing initial conditions involving faster than light velocities or insulating copper wires. When such variations in conditions are actually realized one can often determine empirically whether some candidate generalization continues to hold under such variations. (This is what Boyle did, when he systematically manipulated the pressure of air, measured the associated volume and found a relatively invariant relationship between the two.) Moreover, to at least some extent inference to what would happen under unrealized variations can be justified as a matter of ordinary inductive extrapolation (or interpolation): seeing that the relation F ¼ -kX describing the relation between the extension of a particular kind of spring and the force it exerts continues to hold for various variations in X, one infers that it will hold for intermediate unrealized values of X and perhaps for small increases in X beyond those measured. More generally, whether or not laws of nature are among the truth conditions for counterfactuals, it is not true that in all cases one must know what those laws are to evaluate counterfactuals. If I want to know whether it is true that if I were to drop this rock it would fall to the ground, I don’t need to know the laws governing gravitation or freely falling bodies. Instead I can just drop the rock, taking care to avoid possible confounding factors and observe what happens—the world, rather than elaborate inferences involving possible worlds, gives us the answer. It is bizarre to suppose that I cannot assess counterfactual claims in cases like this or do not understand what they mean without having prior knowledge of laws of nature or a detailed treatment in terms of Lewisian semantics. A similar point often holds for the counterfactuals involved in assessments of invariance. What about the idea that, nonetheless, laws are required to state ‘truth conditions’ for counterfactuals and that this makes the invariance-based account damagingly circular? This complaint only has force if truth conditions for laws can be given in a form that makes no use of counterfactuals.³⁴ That is, the complaint assumes that either (i) law claims can be reduced to claims that do not presuppose counterfactual

³³ This is one reason why I would resist attempts to take a very general and undifferentiated notion of counterfactual dependence as primitive and then to use this to explicate the notion of law, as in Lange (2009b). It is only certain kinds of counterfactuals that are relevant to assessments of invariance. ³⁴ Some will think that either laws must be ‘prior’ to counterfactuals (and hence suitable for explicating them) or counterfactuals must be ‘prior’ to laws. I see no reason to suppose that one of these alternatives ‘must’ be true, independently of an exhibition of the explication in question. Perhaps neither is prior to the other.

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or other modal notions, as in the BSA or (ii) law claims can be elucidated by appealing to special entities like dispositions which are somehow ‘prior’ to counterfactuals. I deny both (i) and (ii)—see Sections 7 and 8.

5. Explanation I suggested above that an important part of understanding the law/initial condition distinction involves an appreciation of what the distinction is to be used for—what purpose(s) or function is served by the distinction. There are many interrelated candidates for such purposes. To begin with, if we can identify a generalization that holds across a range of variations in initial conditions, we can ‘export’ it to different situations within this range and use it to make correct predictions about what will happen in those situations. More generally, satisfaction of the independence conditions of the sort described above will often lead to the elimination of redundancy and unnecessary complexity, as when we replace a number of different local generalizations each holding only in a limited domain with a single overarching generalization which is invariant across all those domains.³⁵ (Think of informational dependence or at least unexplained informational dependence as a kind of redundancy.) Relatedly, as suggested above, making a split between laws and initial conditions allows for the segregation of the disorderly part of what we observe in information about initial conditions, allowing for the formulation of orderly laws. In addition to this, however, it is also natural to think of the law/initial condition distinction as functioning in the service of a certain regulative ideal for explanation and causal analysis. According to this ideal³⁶ explanation and causal analysis often proceed by providing answers to a range of what-if-things-had-been-different questions—that is, by showing how the behavior of the system of kind S would change, under changes in the initial conditions characterizing S, given one or more generalizations applying to S that are invariant under these changes. In this way, we come to see how the behavior of S depends on these initial conditions and how these initial conditions ‘make a difference’ for the behavior of S. The connection with my previous discussion should be obvious: realizing this ideal of explanation requires that we effect a split between laws and initial conditions, with the former invariant over changes in the latter.³⁷ Extending this further, one may also think of the independence conditions imposed on initial conditions as also having a natural motivation in terms of a ³⁵ That is, when a generalization holds only in a limited domain, we may think of this as a kind of failure of complete independence between the generalization and the initial conditions under which it holds. ³⁶ Described in more detail in Woodward (2003). ³⁷ This suggests the following question for those who advocate replacing law-talk laws with something else such as disposition talk: Can such replacements accomplish the same goals as law talk and its accompanying law/initial condition split? I’m inclined to think not, but this question seems to go unaddressed in the dispositions literature.

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related ideal of explanation. Explanations that make use of highly structured initial conditions or correlations among initial conditions or highly unusual initial conditions with no further account of where these come from are generally not regarded as fully satisfying—for the obvious reason that they seem to raise but leave unanswered the question of where that structure comes from (cf. my remarks on the principle of the common cause above). By contrast, explanations that appeal to ‘random’ or relatively unstructured initial conditions do not seem to raise corresponding questions with quite the same urgency—random, uncorrelated, and unstructured initial conditions are a natural stopping place in explanations.³⁸ A common complaint directed against ‘regularity’ accounts of laws, including the BSA, is that if laws are mere summaries of what happens, it is hard to see how they can figure in explanations. The evaluation of this complaint is a complicated matter, but it is worth noting that the invariance-based account is not subject to it. On that account, laws do not just record regularities—instead they encode information about invariance/independence properties and relations. These fit naturally into an account of explanation like that described above that does not take explanation to consist merely in a demonstration that various explananda ‘instantiate’ regularities.

6. Invariance and Independence Extended I have advocated a common framework for characterizing both the relationship between laws and initial conditions and initial conditions themselves that appeals to notions of ‘independence.’ In this section I want to briefly describe several ways in which this common element might be extended and developed.

6.1 Independence Relations among Variables Laws often describe relationships between the values taken by one variable (often but not always placed on the left hand side of an equation) and several other ‘independent’ variables (placed on the right hand side). Even though the equation may not explicitly say this, it is often presumed that these right hand side variables can vary in value independently of each other. For example, in connection with the Newtonian inverse square law, F ¼ Gm1 m2 =r2 , it is usually assumed that, as far as this equation taken in itself goes, the distance r between the two masses m₁ and m₂ can change independently of the values taken by those masses and similarly the masses can change in magnitude independently of each other. In the Lorentz force law, F ¼ qvxB, the charge q of the moving body can vary independently of the intensity ³⁸ None of this should be read as an endorsement of what is sometimes called ‘inference to the best explanation.’ Instead my picture is this: We look for theories and models that exhibit the kind of structure described above because we value explanation as a goal. But the fact that such theories/models would if true provide good explanations is not automatically a reason for taking them to be true. For that we require independent evidence (independent of judgments of explanatory power) for taking the theories/models to be true.

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of the magnetic field, and of its velocity. This independence feature is closely linked to ideas about the possibility of independent variation in initial conditions mentioned in Section 2. Note also this is a feature that is not captured by the standard (1) ‘all As are Bs’ or (2) ‘all As and Cs are Bs’ representations of laws (or, for that matter, the claim that laws represent or are ‘regularities’). In (1) there is only one rhs variable and in (2), there is no representation of whether A and C can vary independently. This is another illustration of the complex and variegated independence information carried by laws which is missed in many philosophical treatments.

6.2 Independence Relations among Laws Marc Lange,³⁹ among others, has drawn attention to the way in which scientists often reason about laws in ways that appear to embody assumptions about the independence of individual laws from one another.⁴⁰ For example, an investigator may ask what the motion of a body would be if it were subject to a gravitational force obeying an inverse cube rather than inverse square law. Such investigations seem to assume that the laws of motion describing how this body responds to forces (e.g. F ¼ ma) are independent of the specific law describing the force itself, so that one can coherently assume a force law different from the actual one, while also assuming the laws of motion remain as they actually are, and use these together to calculate the resulting trajectory.⁴¹ Newton used reasoning of this sort to argue that the evidence supported his inverse square law over alternative gravitational force laws with different exponents. Again, independence in this context might naturally be understood as involving a kind of informational independence, in the sense that alternative gravitational force laws are not inconsistent with the laws of motion (that is, do not imply alternative laws of motion).

6.3 Independence Assumptions in Causal Inference Although our topic is laws rather than causal claims, there are very interesting applications from the machine-learning literature⁴² illustrating the power of the independence-/invariance-based ideas described above, particularly in connection with determining the direction of causation from statistical information. These provide concrete illustrations of what it might mean for initial conditions to fail to be informationally independent of one another and of the ‘laws/causal ³⁹ E.g. Lange (2009b). ⁴⁰ This is my formulation. Lange may not agree with this way of putting things. ⁴¹ If I have understood him correctly, Lange gives this reasoning a very strong modal interpretation: He takes it to involve commitment to the idea that (i) nature is such that if the inverse square law had not held, Newton’s laws of motion still would have held. It seems to me that a weaker interpretation (advocated above) will often suffice to make sense of such reasoning: there is no inconsistency or incoherence in combining the laws of motion with an alternative to the inverse square law. This seems to require a kind of informational independence between the laws, but perhaps not the idea that there is some structure in nature that ensures the truth of the counterfactual (i). ⁴² E.g. Janzing et al. (2012).

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generalizations’ governing a system. Suppose first that X and Y are random variables whose values we observe. Suppose also (i) Y can be written as a function of X plus an additive error term U that is probabilistically independent of X : Y ¼ f ðXÞ þ U with X_|_U (where _|_ means probabilistic independence). Then it can be shown that if the distribution is non-Gaussian, there is no such additive error model from Y to X— that is, no model in which (ii) X can be written as X ¼ gðYÞ þ V with Y_|_V. When applied to empirical data in which the correct causal direction is independently known, the assumption that the correct causal direction is the one in which the cause is independent of the error leads to correct results a majority of the time. One can think of this inference procedure as making use of default assumptions about the independence of incoming influences or initial conditions of the sort described in Section 2. The reliability of the results in this very different context illustrates both the power and generality of such assumptions. Remarkably even in cases in which there are just two variables—X and Y—which are deterministically related via an invertible function (i.e., there is no error term U), it is also possible to determine causal direction by making use of a kind of analog of the idea that laws should be (informationally) independent of initial conditions as described in Section 2. Very roughly, the idea is that if the causal direction runs as X!Y, then we should expect that the function f describing this relationship to be informationally independent of the description of the (marginal) distribution of X (which corresponds in this context to an initial condition)—independent in the sense that knowing this distribution will provide no non-generic information about the functional relationship between X and Y and vice versa. By contrast, it is possible to show that for ‘most’ functional relations g when (X !Y) is the correct direction, writing X as function g of Y (X ¼ g(Y)) will result in a g that is not informationally independent of the distribution of Y. The relevant notion of independence can be understood in terms of algorithmic information theory or in terms of the absence of terms in f that are finely tuned to the distribution of X and the presence of corresponding terms tuned to the distribution of Y in g. Again the procedure is relatively reliable as an empirical matter. This illustrates how looking for representations in which lawful (and causal) relationships are independent of or invariant across initial conditions, with independence understood as informational independence, is a fruitful one, which is at work both in contexts in which physical laws are deployed and in the causal inference contexts described immediately above. This suggests a further speculation: perhaps an alternative to explicating the notion of invariance (and ‘law’ and ‘cause’) in terms of counterfactuals is to employ instead some appropriately behaved notion of informational independence—perhaps at bottom these are alternative ways of getting at the same thing. According to this conception, one looks for a formulation of laws and initial conditions such that the initial conditions are (informationally) independent of one another, the laws are independent of the initial conditions, and the laws are independent of one another. Perhaps such a framework, if it could be developed, might be appealing to those

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who are uncomfortable with the use of counterfactuals in explicating the notion of invariance.

7. Comparison with Alternatives: BSA I turn next to a discussion of the relationship between the invariance-based account and alternatives, beginning in this section with the BSA. I have criticized this account elsewhere.⁴³ Rather than repeating those criticisms, I want to proceed (somewhat) more constructively. From the perspective of the invariance-based account, the key question is whether the resources that are employed in the BSA—the information in the Humean basis, and the notion of a trade-off between simplicity and strength, can be used to (reductively) capture notions like invariance. To adopt assumptions most favorable to the BSA, suppose we had a clear account of the various notions (simplicity, strength, and a best trade-off between these) that go into the BSA and that in our universe there is a single best systemization. What will be the relationship between the axioms and theorems picked out in the BSA framework as ‘laws’ and those claims picked out as laws in the invariance-based account? One possibility is that the two accounts largely agree: the axioms/theorems of the BSA turn out to largely coincide with those generalizations that are invariant under variations in initial conditions when initial conditions meet the independence constraints described above. This would be a happy outcome; presumably the plausibility of each account would be increased by this sort of agreement. Moreover, it would also suggest that the invariance-based account might be reinterpreted so as to meet reductivist requirements, assuming that the BSA does. The other possible outcome is that the two accounts diverge in their judgments of lawfulness, which presumably would mean that the resources of the BSA don’t capture invariance-based notions. While I would not claim that this would show the BSA is mistaken, I believe it would put some pressure on that account, given that the invariance-based account appears to have substantial roots in scientific practice. If, in the case of divergence, BSA advocates claim that it is the invariancebased account that is misguided, what would be the basis for such a claim? Without trying to settle the issue of whether the two accounts agree in their assessments of lawfulness, let me note several points. The first is that, conceptually, the notion of a generalization being an axiom or theorem in a best balanced systemization by no means obviously coincides with the notion of a generalization being invariant in the manner described above. Prima facie, it looks as though a generalization might be ‘simple’ in some relevant sense (or figure as an axiom in some simple systemization) and such that one could derive a lot from it (thus strong

⁴³ E.g. Woodward (2013).

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in some sense) and yet be relatively non-invariant.⁴⁴ Perhaps some cosmological generalizations have this character—assumptions about the homogeneity or isotropy of the universe on a large scale might be part of a simple systemization and might enable one to derive a lot when conjoined with other candidate laws but might nonetheless be non-invariant (would have failed to hold if initial conditions in the early universe had been different). To the extent this is so, the BSA will fail to capture invariance-based notions.

8. Comparison: Special Entities Although the invariance-based account appears to diverge from the BSA in the ways described, it agrees with the BSA in rejecting treatments that appeal to metaphysically special ‘modal’ or ‘non-Humean’ entities (or properties or relations). From an invariance-based perspective, what is problematic about such treatments is not their invocation of modality per se (after all, invariance is naturally understood as a modal notion) but rather their ‘reification’ or ‘ontologizing’ of modal claims—their treatment of modal claims as (or as made true by) existence claims about special entities, with such existence claims having some kind of non-trivial explanatory or ‘grounding’ role for modal claims. I see such accounts as subject to the following dilemma. Suppose, on the one hand, the accounts have no additional physical consequences for what laws are like beyond what is suggested by the invariance-based account. In this case, one worries that the accounts are mere redescriptions of invariance claims in a metaphysical vocabulary. For example, if it is claimed that laws are made true by relations of necessitation between universals, then, assuming that the invariance account is correct as far as it goes, such necessitation relations will need to endow laws with various invariance properties—from the truth of N(F,G) in present circumstances it somehow has to follow that, e.g., All Fs would continue to be Gs if initial and background conditions were to be different in various ways and so on. If this connection with invariance exhausts the physical/empirical content of what N(F,G) involves, then it looks as though we have a mere redescription of these invariance features in the more metaphysical language of necessitation. Similarly for the invocation of dispositions as truth makers: one can build it into one’s conception of a disposition that for aspirin to have a disposition to relieve headaches, aspirin must manifest that disposition (under appropriate triggering conditions) under a range of different circumstances, but this sounds very close to building invariance-linked requirements into

⁴⁴ This outcome (simple and strong generalizations failing to coincide with invariant ones) seems particularly likely if the relevant notion of simplicity is understood, as it often is in discussions of the BSA, purely syntactically—a disjunct is less simple than either of its disjuncts, etc. By contrast, whether a generalization is invariant is not something that can be identified through its syntax.

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the notion of a disposition. In this case there does not seem enough distance between the two for the attribution of the disposition to ‘explain’ the invariance claim. The other possibility is that the metaphysical entity accounts do have additional physical implications that go beyond what is claimed by the invariance account, either by excluding possibilities that the invariance-based account permits or by adding further physical consequences to that account. Either result seems unwelcome since these additional consequences seemed to be reached on non-empirical grounds. As an illustration of the first alternative, many defenders and critics of dispositionbased accounts agree that such accounts have difficulties capturing certain kinds of laws or features of laws—for example, symmetry principles and conservation laws. Apparently the problem is that dispositions are most naturally ascribed to objects, and it does not seem natural to regard, e.g., the conservation of charge as the manifestation of a disposition attributable to any particular object. This leads one leading advocate⁴⁵ to the conclusion that the disposition-based account results in the judgment that symmetry and conservation principles are ‘pseudo-laws’ that may turn out to be ‘eliminated’ as ‘being features of our form of representation rather than features of the world requiring to be accommodated within our metaphysics.’⁴⁶ This judgment/prognostication is certainly at odds with the deep significance attributed to such principles by most physicists and looks like a clear case in which restrictions on the content of science are being motivated by appeals to metaphysical considerations. By contrast, an invariance-based account provides a natural and straightforward treatment according to which conservation and symmetry principles are regarded as genuine laws.

9. Conclusion The upshot of my discussion so far may seem unsettling. I have rejected both reductivist accounts of laws and accounts that appeal to both special metaphysical entities. Putting aside accepting laws as ‘primitive,’⁴⁷ the conventional wisdom is that these accounts exhaust the possible alternative positions. In this concluding section I want to briefly sketch another possibility. This will also constitute a further response to the complaint the invariance-based account fails to provide ‘metaphysical underpinnings.’ Consider, as an analogy, claims about chances when attributed to macroscopic gambling devices, such as roulette wheels, which we can treat as accurately described by deterministic laws, as in (9.3)

The chance of red on the next spin is 0.5.

⁴⁵ Bird (2007). ⁴⁶ Bird (2007, 214). ⁴⁷ As in Maudlin (2007). I regret that I do not have space to discuss this alternative.

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Although such devices are accurately described by deterministic laws, they generate stable relative frequencies of outcomes. These are frequencies that the croupier cannot alter by manipulations that are accessible to her. Moreover, those who have access only to macroscopic information cannot do any better in prediction than by making use of information about such frequencies. A considerable physical/mathematical literature, going back to (at least) Poincaré on ‘the method of arbitrary functions’ explains these features. Very roughly, one can show that (9.4) For a range of different possible dynamics and a large range of possible distributions of initial conditions meeting very generic constraints including the presence of certain symmetries (constraints that a macroscopic croupier will be unable to violate), stable frequencies for device outcomes result. In effect one shows that such frequencies are invariant under manipulations that the croupier is able to perform and under many other changes in the state of the wheel or even its dynamics—in this sense explaining or elucidating why we see stable frequencies. We can thus think of (9.4) as describing those features ‘in the world’ that ‘support’ the ascription of chances to the systems in question and, in some perfectly good sense, ‘explain’ its chancy behavior. Note, however, this does not require that there be some discrete, isolable entity or property with a mysterious ontological status corresponding to ‘chance’ that provides the ‘metaphysical underpinnings’ for the chancy behavior of such systems. The explanation for such behavior instead just involves the diffuse, distributed physical considerations described above. Moreover, if we want to understand the behavior of such systems it would not be fruitful to attempt to provide a ‘reduction’ of chance to something else—e.g., relative frequencies (or some measure of ‘fit’ to relative frequencies) plus additional factors like ‘simplicity.’ Such a reduction, even if it could be provided, would tell us nothing about why such systems exhibit the behavior they do. Again, the explanation for this behavior lies in facts about the relative insensitivity of the behavior of the systems in question to the details of the laws characterizing their dynamical behavior and details of the particular initial conditions induced by the croupier on successive spins. Neither reductivist treatments of chance nor the introduction of chance into our fundamental ontology gives us any insight into such facts. Note that the problem with the latter approach is not that it is false that the device exhibits chance behavior; it is rather that ontologizing ‘chance’ wrongly suggests that there is a special kind of discrete thing or property that (metaphysically) explains this chance behavior, when the actual explanation has to do with the diffuse considerations described above. Finally, note that if one is puzzled about chance and its ascription to gambling devices, much more is relevant than the physical story provided above. Considerations having to do with chance also play distinctive architectural or design roles in our reasoning and decision making—these are matters studied in statistics and

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decision theory. Understanding chance is a matter of understanding both these architectural considerations and the various ways in which chance ascriptions can have physical underpinnings/explanations. Although the analogy is inexact, I suggest that there are a number of parallels with the notion of law. At least in the case of non-fundamental laws, we don’t have to take their invariance properties as brute or primitive. Instead, there is an important scientific project of explaining why various relationships we find in nature are invariant to the extent that they are. The goal of such explanations is to explain why certain kinds of variations in the values of certain variables do not matter for why the relationships hold—why the relationships are stable across variations in those variables. Similarly to the explanations of the behavior of gambling devices, such explanations typically take the form of showing that provided systems subject to such laws satisfy certain very generic constraints, further variations in other variables will not affect whether they exhibit certain stable patterns of behavior. Details vary across cases (and in many cases we do not presently know what is responsible for the invariant relationships we see), but examples include explanations in terms of statistical mechanics for why variations in molecular details that are consistent with certain macroscopic constraints do not matter for whether various laws of thermodynamics hold, explanations of various aspects of critical point behavior in terms of the renormalization group, and demonstrations in particle physics in the form of ‘decoupling theorems’ showing that, as long as generic constraints are met, the detailed structure of presently unknown high-energy theories is irrelevant to laws at lower energy scales. Several points about such explanations deserve emphasis. Note first that the explananda are not (or not just) generalizations specifying that certain regularities hold. Instead the explananda are facts about the invariance of various relationships. This involves a different kind of explanation than a mere derivation showing that some generalization is true—instead we are looking for an explanation of the generalization’s stability. As with chance, such explanations will appeal to complex and distributed considerations and to generic constraints rather than reductions or special entities and properties. Indeed, it is hard to see how any of the latter could explain facts about invariance any more than ‘chance’ is the name of something that explains the behavior of roulette wheels. My suggestion is that insofar as explanations of (or underpinnings for) invariance in terms of what is in nature are possible, what we should be looking for are explanations that have the character described above. I concede that this strategy does not help us to understand the invariance of laws, if any, that are ultimate or fundamental in the sense of having no further explanation, but at this point in the development of science it is not clear what would provide illumination about this. Note also that, on this view, just as the case with ‘chance’ what is wrong with the postulation of special entities to serve as metaphysical grounds or truth makers for laws of nature is the reification and illusion of explanation this involves. It is not that it is false that aspirin has the power to relieve

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headaches, rather the point is the ascription of this power to aspirin does not explain its behavior.⁴⁸ Finally, let me add that just as in the case of chance, there is more to be said about laws than what in nature ‘underpins’ them. An important part of the project of understanding the notion of law involves understanding the role this notion plays in our reasoning and decision making, what sort of evidence is appropriate for establishing law claims, how this notion connects to other scientifically important notions, and so on. These fall into the general category of architectural/design considerations. They connect with the methodological role played by laws. A ‘metaphysics’ of laws will not tell us about these matters. I have tried to provide some preliminary suggestions above.

⁴⁸ There is a tendency in philosophical discussion to describe views that deny that there are special metaphysical entities that serve as truth makers for laws as ‘anti-realist.’ This strikes me as tendentious. If realism about laws is just the thesis that claims about laws can be true or false, my view of laws is a ‘realist’ one, even though it rejects ‘special entity’ accounts. We need to separate the contention that law claims can be true or false from the contention that it is illuminating to think of such claims as made true by special entities like powers or dispositions.

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10 How the Explanations of Natural Laws Make Some Reducible Physical Properties Natural and Explanatorily Powerful Marc Lange

1. Introduction It has often been recognized that (in Lewis’s words) ‘the scientific investigation of laws and of natural properties is a package deal . . . physicists posit natural properties such as the quark colours in order to posit the laws in which those properties figure, so that laws and natural properties get discovered together.’¹ As Lewis’s example of quark colors exemplifies, when philosophers study the entanglement between laws of nature, scientific explanation, and the naturalness of certain properties, the focus is often on fundamental laws and fundamental properties. Indeed, Lewis regards any reducible (and therefore non-fundamental) physical property as ineligible to be perfectly natural. However (I will argue), scientific practice recognizes certain reducible physical properties (such as the property of having a given center of mass and the property of having a given Reynolds number) as genuine, explanatorily potent respects in which various systems are alike. Among the non-fundamental physical properties, how are those that are natural and explanatorily potent set apart from those that are arbitrary algebraic combinations of more fundamental properties?² This is the question that I will motivate, explore, and try to answer in this chapter. Let’s start with an example. Suppose that a seesaw on earth has two point bodies sitting on it: one on the right, with mass mR ¼ 150 kg, sitting xR ¼ 2 m to the right of ¹ Lewis (1983, 368). ² In putting my question on the table, I will treat this distinction as if it were sharp. In Section 6, however, I will remove this oversimplification. Of course, this question presupposes that a natural property does not have to be fundamental; I will return to this point.

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the fulcrum, and one on the left, with mass mL ¼ 200 kg, sitting xL ¼ 1 m to the left of the fulcrum. Why does the seesaw tip to the right? This question asks for an event’s scientific explanation. According to classical physics (a qualification that applies to all of my examples), one explanation of the event is that it is a natural law that any such seesaw³ tips to the right if mR xR  mL xL > 0, and for the given seesaw, mR xR  mL xL ¼ 100 kg-m. In this explanation, the crucial feature of the seesaw system is that its mR xR  mL xL ¼ 100 kg-m. Of course, this property is simply certain of the system’s more basic properties standing in a certain relation; the system’s having mR xR  mL xL ¼ 100 kg-m is not a fundamental property, but rather is reducible to its having mR ¼ 150 kg, xR ¼ 2 m, mL ¼ 200 kg, and xL ¼ 1 m. Despite its reducibility, the system’s having mR xR  mL xL equal to 100 kg-m can help to explain the seesaw’s tipping to the right. We are not inclined to say that since mR xR  mL xL ¼ 100 kg-m is nothing more than a relation among mR, xR, mL, and xL, the outcome is not explained by the system’s mR xR  mL xL (but instead by its having those more basic properties). Some philosophers may insist that the seesaw tips to the right not because its mR xR  mL xL ¼ 100 kg-m, but rather because its mR xR  mL xL > 0. For my purposes, it does not matter which of these views is correct. Perhaps each of these properties of the seesaw system can explain why it tips to the right. The important point is that at least one of them has explanatory power despite being reducible. I shall refer to such a property as ‘the seesaw’s mR xR  mL xL .’ There is presumably yet another way to explain the same event: the seesaw tips to the right because its mR xR ¼ 300 kg-m, its mL xL ¼ 200 kg-m, and it is a natural law that any such seesaw tips to the right if it has mR xR > mL xL . Once again, the seesaw’s having mR xR ¼ 300 kg-m is reducible to its having mR ¼ 150 kg and xR ¼ 2 m, and its having mL xL ¼ 200 kg-m is likewise reducible. But their reducibility does not deprive them of explanatory power. Not every reducible physical property is like these properties in possessing explanatory power. Suppose two point bodies have long been held at rest, 1 cm apart. The magnitude of each body’s electrostatic force (Fe) on the other is explained by the bodies’ charges (q₁, q₂) and separation (r), together with Coulomb’s law ðFe ¼ q1 q2 =r2 Þ. Likewise, the magnitude of the bodies’ mutual gravitational attraction (Fg) is explained by the bodies’ masses (m₁, m₂) and separation (r), together with Newton’s gravitational force law ðFg ¼ Gm1 m2 =r2 Þ and the value of the gravitational constant (G). Suppose that the pair of bodies possesses the property of having m1 m2 =r2 ¼ 30;000 kg2 =m2 . This property does not help to explain the mutual gravitational force, though it combines with Newton’s gravitational force law and G’s value to entail that force. Similarly, the pair’s q1 =r and q2 =r do not join Coulomb’s law ³ That is, one with a rigid arm to which the two point bodies are secured, the bodies feeling no influences besides the contact force from the arm and a force from an ambient gravitational field that can safely be treated as approximately uniform over the seesaw.

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in explaining the electrostatic force (though they suffice with the law to entail that force). The same applies to other arbitrary combinations, such as the pair’s q1 r=q2 and q2 2 =r3 (the product of which is again q1 q2 =r2 ). These two properties are manufactured clusters of quantities—mere conglomerates of certain properties that can actually do the explaining. There is a distinction here. That q₁q₂ (measured in esu²) exceeds r² (in cm²) does not join with Coulomb’s law to explain why the electrostatic force exceeds 1 dyne, whereas that mR xR exceeds mL xL joins with the relevant law to explain why the seesaw tips to the left. The pair’s having m1 m2 =r2 ¼ 30;000 kg2 =m2 has no explanatory significance whereas the seesaw’s mR xR  mL xL has explanatory power. What is responsible for this distinction among the reducible physical properties, making some but not others explanatorily powerful? In this chapter, I examine this question and try to answer it. Of course, the seesaw’s having mR xR  mL xL > 0 is logically equivalent to its having xR  mL xL =mR > 0. Yet its xR  mL xL =mR is just another arbitrary combination having no explanatory significance. How can mR xR  mL xL > 0 help to explain the outcome when it is logically equivalent to xR  mL xL =mR > 0, which is explanatorily powerless? This challenge is based on the mistaken presupposition that if p and q are logically equivalent, then they must be equivalent in their explanatory power. As a simple counterexample, consider that Coulomb’s law (l) together with various initial conditions (c) explains why the two point bodies in my earlier example feel their mutual electrostatic force (f ). Since l&c logically entails f, l&c is logically equivalent to l&c&f. Yet l&c&f fails to explain f. There are obviously many such tricks: l&(c or ~f )&f is also logically equivalent to l&c but lacks its explanatory power regarding f. Two bodies of liquid fit exactly into the same glass because they have the same volume V, not because they have the same V² or the same 1/V.⁴

⁴ Perhaps ‘l&c&f ’ denotes the same fact as ‘l&c.’ (I will have nothing to say about fact individuation.) Nevertheless, ‘l&c&f explains why f ’ is not equivalent to ‘l&c explains why f.’ For instance, if causal explanation is understood to be the kind of explanation at issue, then ‘l&c&f explains why f ’ holds exactly when ‘l&c&f ’ captures all and only the contextually relevant features of f ’s causal history (or, more broadly, of the world’s network of causal relations). But ‘l&c&f ’ cannot do this because f is not part of its own causal history. (In different contexts, different features of f ’s causal history—more distal or proximate, more coarse-grained or fine-grained, . . . —will be of interest, but there is no context in which f is a feature of interest because f is not part of its own causal history.) Analogous considerations apply to other varieties of scientific explanation. For instance, consider an ‘identity explanation’ from Achinstein (1983, 236): ‘The reason that ice is water is that ice is composed of H₂O molecules’ is an explanation whereas (Achinstein notes) ‘The reason that ice is water is that ice is water’ is not an explanation, even though the property of being water and the property of being composed of H₂O molecules ‘are clearly identical’ (Achinstein 1983, 236). Even if explanation is a relation among facts, evidently it must also be sensitive to the ways in which those facts are expressed. This is an old point; as Mackie (1974, 260) said, ‘Oedipus’ having married Jocasta is the same fact as his having married his mother, and if the latter caused the tragedy, so did the former. Yet the latter description helps to explain the tragedy in a way that the former does not’ (cf. Davidson 1967, 695).

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In asking what makes the seesaw’s mR xR  mL xL differ in explanatory power from its xR  mL xL =mR , I am asking about a contrast between two reducible physical properties. Neither is a multiply realizable property that is irreducible to more basic properties in view of being the property of playing a certain functional or causal role, such as the property of being a predator, being a gene,⁵ or being money.⁶ Of course, to say that the properties with which I am concerned are reducible is not to say that they are eliminable; among the seesaw’s properties is the property of having xR  mL xL =mR > 0. Of course, the seesaw would still have tipped to the right even if mR, xR, mL, and xL had been different, as long as mR xR  mL xL > 0 had still held. But this is not enough to distinguish mR xR  mL xL from xR  mL xL =mR since the seesaw would still have tipped to the right even if mR, xR, mL, and xL had been different, as long as xR  mL xL =mR > 0 had still held—or, equivalently, as long as mR =xL > mL =xR had still held—and yet all of these reducible properties are arbitrary combinations having no explanatory significance. In asking what makes the seesaw’s mR xR  mL xL differ in explanatory power from its xR  mL xL =mR , I am not asking what makes one rather than the other a cause of the seesaw’s tipping to the right. For one thing, causes are events and it is not obvious that the seesaw’s being mR xR  mL xL > 0 is an event at all (even if, e.g., its having mR ¼ 150 kg is an event). Perhaps it seems more like a mathematical relation’s holding among events. For another thing, many non-causes (such as absences, omissions, and laws) figure in causal explanations, and furthermore some scientific explanations are non-causal in that they do not explain by describing the world’s causal nexus.⁷ I am concerned directly with scientific explanations, not with causal relations. Let’s look at an example. Consider two simple pendulums on earth, having lengths l₁ ¼ 1 m and l₂ ¼ 0.5 m and periods T₁ ¼ 2.0 s and T₂ ¼ 1.4 s. That l₁ > l₂ (together with various laws and background conditions) explains why T1 > T2 . Though this is a causal explanation, there may well be no event consisting of l₁ exceeding l₂; there may merely be separate events involving each of the two pendulums. (The view that the difference between l₁ and l₂ is metaphysically ineligible to be a cause is like Strevens’s view regarding the difference in metabolic efficiency between two strains of bacteria, living in separate petri dishes: ‘Metaphysically, this difference is incapable of supplying causal oomph in the case at hand, being as it is an abstract relation between two states of affairs that act independently.’)⁸ In virtue of what, then, does the relation l₁ > l₂ (together with various laws and background conditions) causally explain why T1 > T2 ? Plausibly, the l-relation causally explains the T-relation because by law, the l-relation entails the T-relation, and because each l-relatum (together with laws and background conditions) causally explains the corresponding T-relatum. That is:

⁵ Kitcher (1984).

⁶ Fodor (1974).

⁷ Lange (2013).

⁸ Strevens (2008, 173–4).

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(1) l1 ¼ 1 m (together with background conditions) and the law that T ¼ 2π√ðl=gÞ causally explains why T1 ¼ 2:0 s. (2) l2 ¼ 0:5 m (together with background conditions) and the law that T ¼ 2π√ðl=gÞ causally explains why T2 ¼ 1:4 s. (3) By law (under the background conditions), if l1 > l2 , then T1 > T2 . Thus, l1 > l2 (together with various laws and background conditions) causally explain why T₁ > T₂ even if l1 > l2 is, strictly speaking, not a cause. Does an analogous account reveal what makes the seesaw’s mR xR  mL xL (together with various laws and background conditions) able to explain its tipping to the right? The torque τR on the right (tending to rotate the seesaw clockwise) is causally explained by xR, the downward force FR on the right, and the law τR ¼ xR  FR . The force FR, in turn, is causally explained by mR, g, and the law FR ¼ mR g. The seesaw’s left side receives similar treatment. From the law τ ¼ Iα relating net torque, the seesaw’s moment of inertia I, and its angular acceleration α, it follows that if τR > τL , then the seesaw tips to the right. Thus, by analogy to the way that the pendulum’s l1 > l2 helps to causally explain why T1 > T2 , we have (1΄ ) mR ¼ 150 kg and xR ¼ 2 m (together with background conditions and laws) causally explains why τR ¼ 2940 N-m. (2΄ ) mL ¼ 200 kg and xL ¼ 1 m (together with background conditions and laws) causally explains why τL ¼ 1960 N-m. (3΄ ) By law (under the background conditions), if mR xR > mL xL , then τR > τL and so the seesaw tips to the right. Do we have here an account of why the seesaw’s mR xR  mL xL , despite being reducible, manages (together with various laws and background conditions) to explain why the seesaw tips to the right? No. The explanans in the pendulum example is that l1 > l2 (together with various laws and background conditions), and this relation accordingly appears in (3) with its two relata figuring in (1) and (2), respectively. However, (3΄ ) includes mR xR > mL xL , whereas our aim was to account for the explanatory power of mR xR  mL xL > 0. Of course, these two inequalities are logically equivalent. But we cannot thereby argue that they are equivalent in explanatory power, on pain of taking the seesaw’s xR  mL xL =mR > 0 as also having the power to help explain the outcome. We were trying to identify what makes the seesaw’s mR xR  mL xL differ in explanatory power from its xR  mL xL =mR . Although we were trying to account for the explanatory power of mR xR  mL xL > 0, (3΄) instead includes mR xR > mL xL . Do we have here, then, at least an account of how the seesaw’s mR xR > mL xL (together with various laws and background conditions) explains why the seesaw tips to the right? That would still be some progress since we are also trying to understand why the seesaw’s mR xR and mL xL have explanatory power whereas its mL xL =mR , for instance, does not. However, we have not gotten even

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that far. In the pendulum example, the relation l1 > l2 doing the explaining appears in (3) and the two relata (l₁ and l₂) figure in (1) and (2), respectively. But although mR xR > mL xL appears in (3΄), mR xR and mL xL do not figure in (1΄ ) and (2 ΄ ), respectively. Rather, mR and xR figure individually in (1΄) and mL and xL figure individually in (2 ΄ ). We should not underestimate the difference that this makes. To replace (1΄ ) mR ¼ 150 kg and xR ¼ 2 m (together with background conditions and laws) causally explains why τR ¼ 2940 N-m with (1΄΄ ) mR xR ¼ 300 kg-m (together with background conditions and laws) causally explains why τR ¼ 2940 N-m would be to presuppose that mR xR is explanatorily powerful despite being reducible. What makes mR xR explanatorily powerful, unlike arbitrary reducible properties such as mL xL =mR , is exactly what we are trying to understand. Notice, by the way, that the seesaw’s mR xR  mL xL does not derive its explanatory power from its being equal to the magnitude of some mechanical cause of the outcome, whether that cause be a force or torque, component or net. The various forces are mR g downward on the right and mL g downward on the left (along with an upward force exerted by the fulcrum), and the net torque is equal to gðmR xR  mL xL Þ. Of course, it follows that ðmR xR  mL xL Þ’s sign determines the net torque’s sign and hence the angular acceleration’s direction. But the same is true of ðxR  mL xL =mR Þ’s sign, for example. In taking the seesaw’s mL xL =mR and other arbitrary combinations as constituting properties, I am employing an ‘abundant’ view of properties⁹ according to which every class of possible entities is associated with a property and every class of n-tuples of possibilia is associated with a relation.¹⁰ We might be inclined to say that what privileges the explanatorily powerful properties over the arbitrary combinations is that they alone are natural (a.k.a. ‘sparse’) properties. Plausibly (as emphasized by Fodor and Loewer),¹¹ the natural properties alone figure in laws of nature, whereas an unnatural property may figure in a logical consequence of laws that, though physically necessary (obviously), is not a law (such as that under certain specified conditions, a seesaw tips to the right if xR  mL xL =mR > 0).¹² If natural laws are crucial ⁹ In the sense of Lewis (1986a, 59). ¹⁰ Oftentimes I use ‘properties’ to include relations. Furthermore, even if it is logically impossible for anything to possess the property of having mR xR  mL xL > 0 but to lack the property of having xR  mL xL =mR > 0, I take these to be distinct properties. Thus I say that a property is associated with a class of possibilia, not that a property is identical to a class of possibilia or that a class of possibilia is associated with exactly one property. ¹¹ Fodor (1974) and Loewer (2007, 316). ¹² For more on the distinction between laws and physically necessary non-laws, see Lange (2000, 201–7).

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to scientific explanations, then a difference in naturalness between the seesaw’s mR xR  mL xL and its xR  mL xL =mR might seem a promising way to account for their difference in explanatory power. To refine this response, we would have to consider whether naturalness comes in degrees, as Lewis believes.¹³ The difference in explanatory power between the seesaw’s mR xR  mL xL and its xR  mL xL =mR does not seem to be a matter of degree; the latter seems explanatorily powerless. (I will return to naturalness as a matter of degree in Section 6.) We would also have to consider whether any reducible properties could be (perfectly) natural. Lewis thinks not, since regarding (perfectly) natural properties, he says that ‘there are only just enough of them to characterise things completely and without redundancy.’¹⁴ But if no reducible property is natural, then naturalness cannot be used to privilege the reducible properties that possess explanatory power. I will set aside both of these concerns, since an appeal to naturalness faces a more fundamental obstacle: it does little to account for the differences in explanatory power among various reducible properties. If the seesaw’s mR xR  mL xL and its xR  mL xL =mR differ in explanatory power because they differ in naturalness, then in virtue of what do they differ in naturalness? Lewis is prepared to take a property’s naturalness as a metaphysical primitive, a brute fact.¹⁵ I am not—or, at least, I am considerably less willing to do so for reducible properties than for fundamental properties such as (perhaps) mass, electric charge, and quark color. Though an appeal to naturalness does little to explain why the seesaw’s mR xR  mL xL differs in explanatory power from its xR  mL xL =mR , I am happy to use naturalness to express our original puzzle in a new way: We want to know what makes certain reducible physical properties but not others natural. Two seesaws being alike in their mR xR  mL xL ‘makes for qualitative similarity’¹⁶ and so can explain why they are alike in both tilting to the right. It is no coincidence that they both tilt to the right; the two seesaws have something in common (namely, having mR xR  mL xL > 0) that explains the common tilt. In contrast, although the same two seesaws also have the same value for xR  mL xL =mR , this is not a genuine respect of similarity, just as two pairs of charged bodies’ having the same value for q1 r=q2 does not amount to their resembling each other. That two systems having a given reducible property in common are genuinely similar in a certain respect is often emphasized by scientists when they introduce a single term for that reducible property or appeal to that property in explanations. For example, when Whewell introduced a single term into English (‘labouring force’— later supplanted by the term ‘work’) for the product of a force and the distance through which it acts, he emphasized that two cases of equal laboring force constitute ‘the same thing’ whatever the particular values of force and distance underwriting ¹³ Lewis (1986a, 61). ¹⁵ Lewis (1983).

¹⁴ Lewis (1986a, 60). ¹⁶ Lewis (1986a, 60).

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them—whether they involve ‘one pound raised twelve feet; two pounds raised six feet; three pounds raised four feet; six pounds raised two feet; and so on.’¹⁷ One advantage of formulating our question in terms of the difference in naturalness between, say, mR xR  mL xL and xR  mL xL =mR is that it emphasizes that to answer our question, we do not have to give necessary and sufficient conditions for one fact or event to explain another. Rather, our question is confined to what makes one property natural enough—far enough from being an arbitrary conglomerate of more fundamental properties—to be eligible to figure in explanations.¹⁸ Whether or not the possession of a certain property that is natural enough to explain actually does help to explain a given fact is another question. Some readers may suspect that despite my suggestive examples, there is no difference in explanatory power between mR xR  mL xL and xR  mL xL =mR ; the latter explains whatever the former does, just not as perspicuously. The best argument for this view, it seems to me, would be the failure to find any good account of why they differ in explanatory power. In this chapter, I aim to offer such an account. In Section 2, I try to motivate further the distinction in which I am interested by contrasting centers of mass with reduced mass; I also refine the question I am addressing by contrasting it with other questions that philosophers have asked regarding centers of mass. In Section 3, I underline the importance of the distinction among reducible physical properties between those that are and those that are not sufficiently natural to be explanatorily powerful. I show how the failure to respect this distinction causes trouble for a recent account of scientific explanation given by Strevens.¹⁹ In Section 4, I argue that in scientific practice, many important dimensionless quantities, despite being reducible, are treated as sufficiently natural to be explanatorily powerful. In Section 5, I offer my account of why certain reducible physical properties but not others are sufficiently natural to be explanatorily powerful, and I apply this account to all of the foregoing examples, along with several others. Roughly speaking, I propose that the possession of some reducible property P is natural enough to explain some fact, once P is joined with a law L in which P figures (and perhaps further conditions), exactly when there is an explanation of L that the expression for P enters as a unit. The alternative to its entering some explanation of L as a unit is its emerging somewhere in the course of every explanation of L by a fortuitous combination of the various properties to which P reduces. I conclude in Section 6. ¹⁷ Whewell (1841, viii). ¹⁸ This formulation acknowledges that naturalness comes in degrees (see Section 6). Once again, however, in using the concept of naturalness to express our question, we must reject the thought that a reducible property cannot be natural (to any appreciate degree) because it is redundant in characterizing the world. Non-redundancy seems to me quite distinct from being a genuine respect of similarity (rather than an arbitrary or gerrymandered ‘property’), figuring in laws (rather than merely in logical consequences of laws), and being explanatorily powerful. Nevertheless, many metaphysicians simply identify the perfectly natural and fundamental properties. ¹⁹ Strevens (2008).

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2. Centers of Mass and Reduced Mass As we have seen, the seesaw’s tipping to the right can be explained in terms of the forces on it and the torques they produce. The expression mR xR  mL xL appears late in this explanation: in the sum of the torques from the left and right sides. This expression appears much earlier in a different (but equally correct) explanation of the seesaw’s behavior, namely, as an instance of the ‘tipping law’: (Tipping law) A rigid, initially stationary body of mass M (small enough that earth’s gravity varies negligibly over the body) under no external forces besides earth’s gravity and some support – tips over so as to lower its center of mass, if its center of mass is not located directly above or below a point of support, and – is at equilibrium otherwise where the body’s having its center of mass at location r is identical to the body’s having ðΣmi ri Þ=M (or, in the case of a continuous mass distribution, R ð ρðrÞrdVÞ=MÞ equal to r. The expression mR xR  mL xL appears as the numerator in the location of the seesaw’s center of mass. If mR xR  mL xL > 0, the seesaw’s center of mass is located to the right of the fulcrum (the point of support), and so by the tipping law, the seesaw tips to the right (thereby lowering its center of mass). Thus, the tipping law explains the seesaw’s behavior. In the seesaw’s having its center of mass at ðmR xR  mL xL Þ=M, we have a reducible property possessing explanatory power. Our challenge is to understand why such a reducible property is natural enough to explain. The center of mass’s role in the tipping law might seem to account somehow for its explanatory power. However, not every mathematical combination of various factors is natural enough to help explain the effect of those factors even if that combination figures in a law determining that effect. Let’s look at a reducible property that is like center of mass in many ways but is not explanatorily relevant. The tipping law says roughly that a rigid, extended body moves in the manner of a single point body located at the extended body’s center of mass. Here is a similar law concerning any classical two-body problem (i.e., any case where two point masses, feeling no external forces, move under their mutual influence): the first body moves relative to the second in the manner of a point body feeling the same force as the first body feels but with mass equal to the pair’s ‘reduced mass’ μ ¼ m1 m2 =ðm1 þ m2 Þ. For example, suppose that each of two mass points feels only the other’s gravity. The above law can be used to predict their relative acceleration a: Treat one body as if it had mass μ and felt the force (given by Newton’s gravitational force law Fg ¼ G m1 m2 =r2 ) it actually feels. By Newton’s second law of motion (force ¼ mass  acceleration)

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G m1 m2 =r2 ¼ μa. Inserting some initial conditions—let’s say: m₁ m₂ ¼ 3 g², r ¼ 1 cm, and μ ¼ 3/4 g—we find a ¼ 2:67  107 cm=s2 . This argument is computationally very efficient and the law it uses is very concise. The concept of reduced mass thus promotes expressive economy and calculational ease, just like the concept of center of mass. But unlike the location of a system’s center of mass, the magnitude of a pair’s reduced mass does not help to explain its behavior. The pair’s μ and m₁m₂ are merely arithmetic combinations of some of the properties actually responsible for the bodies’ relative acceleration: let’s say, m1 ¼ 3 g and m2 ¼ 1 g (which join r, G, and the laws to explain a). Despite its utility, reduced mass is just a convenient conglomerate of some explanatorily powerful properties— and thereby differs from center of mass. My aim is to understand the grounds of this difference. Although we can predict the relative acceleration in a two-body problem by imagining a body of mass μ feeling a given force, obviously the actual system contains no such body. But this fact is not what makes reduced mass explanatorily idle (despite its predictive utility). After all, a body’s center of mass may lie entirely outside that body, where there is nothing. Yet this fact does not keep the center of mass from helping to explain the outcome. A body has the property of having its center of mass at a given location, and a pair of bodies has the property of having a given reduced mass. The question is why these properties differ in their explanatory power, not which of them refers to the location or mass of some additional body existing alongside the bodies in question. Azzouni distinguishes ‘physically significant items which are not physically real (such as centers of mass)’ from ‘mere mathematical objects (such as various functions of physical quantities that we could define).’²⁰ Though Azzouni does not define ‘physically significant,’ his distinction appears not to be the same as the one I am examining. Azzouni characterizes reduced mass as ‘physically significant,’ whereas his example of a quantity that does ‘not correspond to anything physically significant’ is the product m₁m₂ of the masses of two gravitationally interacting bodies.²¹ But as we have just seen, these two quantities both figure in the argument for predicting relative acceleration in a classical two-body problem. If μ’s ‘physical significance’ were associated with the explanatory power of arguments like this, then Azzouni would have to regard m₁m₂ as likewise physically significant, which he does not. Azzouni treats a pair’s center of mass as a hypothetical body that ‘has mass m₁ þ m₂’ and so as a useful fiction.²² By contrast, I see a pair’s having its center of mass at a given location as a property just as real as any other that the pair possesses, though as distinguished in explanatory power even from such scientifically useful properties as having a given reduced mass. Haugeland also considers the ontological ²⁰ Azzouni (1997, 200).

²¹ Azzouni (1997, 200).

²² Azzouni (1997, 199).

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status of a system’s center of mass, but unlike Azzouni, he says that a center of mass ‘is nothing other than a spatial point at a time, or a point trajectory through time. Hence, its ontological status is exactly on a par with that of any other spatial point or trajectory,’ even a scientifically insignificant one.²³ My concern, by contrast, is with the property rather than the point (or trajectory); as a property of a given system, its center of mass differs from many of its other (abundant) properties in being natural enough to play an explanatory role. Haugeland’s remarks are directed against Dennett. Yet unlike Haugeland (but like me), Dennett emphasizes the explanatory role played by centers of mass,²⁴ despite being reducible: Explanations may refer to centers of gravity. Why didn’t the doll tip over? Because its center of gravity was so low.²⁵ What did [America’s Cup skipper] Connor do overnight to cause his boat to be so much faster? He lowered its center of gravity. Of course he did this by moving gear, or adding lead ingots to the bilge, or replacing the mast with a lighter mast, or something—but what caused the boat’s improvement was lowering its center of gravity. This is not just casual shorthand; it is the generalization that explains why any of these various changes (and a zillion others one could describe) would likewise cause an improvement in performance.²⁶

That all of these zillion changes would enhance the boat’s speed is no coincidental similarity among them. Rather, they all have this property because they share the property of lowering the center of mass. Properties involving the center of mass must thus be natural enough to explain. However, Dennett says little about what makes the center of mass natural enough to play this explanatory role. He says only that it is ‘implicated in important causal generalizations.’²⁷ But as we have seen, precisely the same can be said of reduced mass—and, for that matter, of m₁m₂ (since it appears in Newton’s gravitational force law, for example). To account for the differences in explanatory power among various reducible physical properties, it will not suffice to appeal to remarks like Dennett’s: that centers of mass ‘deserve to be taken seriously, learned about, used. If we go so far as to distinguish them as real (contrasting them, perhaps, with those abstract objects which are bogus), that is because we think they serve in perspicuous representations of real forces, “natural” properties, and the like.’²⁸ That there are ‘plenty of true, valuable, empirically testable things one can say with the help of the term’²⁹ may account for the predictive utility and convenience of the concept of

²³ Haugeland (1993, 55). ²⁴ Dennett refers to ‘center of gravity,’ which in a uniform gravitational field is identical to center of mass. ²⁵ Dennett (1983, 380, his emphasis). ²⁶ Dennett (2000, 357–8, his emphasis). ²⁷ Dennett (2000, 357). ²⁸ Dennett (1991, 29, his emphasis). ²⁹ Dennett (1983, 380).

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center of mass, but cannot account for its explanatory power or differentiate it from plenty of reducible properties that lack explanatory power.

3. Reducible Properties on Strevens’s Account of Scientific Explanation On the strength of several examples, I have suggested the importance in scientific practice of the distinction between reducible physical properties that are sufficiently natural to have explanatory power and those that are too unnatural to explain. Before looking further into science, let me give a different argument for this distinction’s importance. I shall give an example of a recently proposed philosophical account of scientific explanation that fails to attend to this distinction and suffers for it. I have in mind Michael Strevens’s proposal in his admirably ambitious book Depth: An Account of Scientific Explanation.³⁰ On Strevens’s account, scientific explanations trace the operation of fundamental causal mechanisms, but an explanation must typically abstract from the explanandum’s specific causes. That is because these details turn out not to ‘make a difference’ to the explanandum and hence to be explanatorily irrelevant. However, Strevens’s account of abstraction ends up treating all manner of conglomerate physical properties as explanatorily powerful when in fact, as we have seen, only a select few are. In the simplest case, Strevens says, an explanandum arises from various causes, laws, and conditions that logically entail it through a deductive argument that ‘mirrors’ the causal process that produces it. An explanandum can have many such ‘causal models,’ and when a model is optimally pruned, according to Strevens, it is transformed into an explanation. Strevens’s basic rationale for excising part of a model is that without it, the model still manages to entail the explanandum. However, as Strevens fully recognizes, this approach encounters two obvious obstacles. Suppose that c explains e. (i) If the model contained (c& a) in place of c, for some arbitrary truth a, then e would still follow by an argument that mirrors the causal process that actually produced e, and generally the new model would no longer entail e if (c & a) were removed from it. But we do not regard (c & a) as explanatorily relevant. Rather, c is relevant and a is irrelevant. Even if a specifies further details of the events given in c, e happens because of c and not because of a. (ii) Suppose that if a given falsehood b replaced c in the model, then e would still follow by an argument that traces a nomologically possible causal process. (Perhaps c and b are entirely dissimilar schemes for generating the same net force.) Then if (c or b) replaced c in the model, e would still follow by an argument that mirrors the causal process that actually produced e, and generally the new model would no longer entail e if (c or b) were removed from it. But we do not regard (c or b) as explanatorily relevant. ³⁰ Strevens (2008).

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Rather, c is relevant. It might be suggested that the reason why c rather than (c&a) explains e is that c is weaker than (c&a) and still entails e. But this suggestion cannot be correct: (c or b) is weaker than c and still entails e, yet c rather than (c or b) explains e. Strevens’s solution to these problems, in brief, is to require that an explanation arise from ‘optimizing’ a causal model. Part of optimizing it involves abstracting: (c&a) must be abstracted to c, since c says less than (c & a) does. (We cannot abstract to a, since the resulting model does not entail e.) However, Strevens says, abstraction must not be allowed to go too far: if c is abstracted to (c or b), then the model’s metaphysically possible realizers may well be ‘incohesive’ in that they fail to form a ‘contiguous set’ in ‘causal similarity space.’ An explanation optimizes gain of abstraction against loss of cohesion. Something like Strevens’s strategy seems like a natural way to face the two obstacles. It deals nicely with Strevens’s example of a cannonball that breaks a window exactly when its momentum (mass  velocity) exceeds 20 kg m/s. Intuitively, neither the ball’s mass of 10 kg nor its speed of 5 m/s qualifies as an explainer, since the threshold involves momentum rather than either mass or velocity by itself. The condition that the ball’s momentum exceeds 20 kg m/s is generated from the model’s undergoing as much abstraction as it can withstand (and involves no incohesion); further abstraction to the ball’s momentum exceeding 10 kg m/s would prevent the model’s entailing the window’s breaking, and further abstraction to more than one projectile would produce incohesion. Strevens’s approach thereby correctly identifies the explanatorily relevant feature. By abstracting mass or velocity separately, we would have gotten the wrong answer. The window would still have broken as long as the speed had exceeded 2 m/s (since the mass would still have been 10 kg, and so the momentum would have exceeded 20 kg m/s). But by treating velocity separately from mass, this approach fails to abstract far enough. However, another case shows that Strevens’s approach cannot pick out the reducible properties that are explanatorily powerful. Suppose that two point bodies charged to +5 statcoulombs and 6 statcoulombs, respectively, and long held at rest, 1 cm apart, are suddenly released in an otherwise empty universe (where other forces between them are negligible). By Coulomb’s law ðFe ¼ q1 q2 =r2 Þ, their mutual attraction measures 30 dynes. What explains this 30-dyne force? Just as Strevens’s approach applied to the cannonball yields an explanation in terms of momentum mv, rather than in terms of mass m or velocity v separately, so the approach applied to this electrostatic case yields an explanation in terms of q1 q2 =r2 , rather than in terms of the individual charges or their separation. That the algebraic quantity q1 q2 =r2 equals 30 dynes is as abstract as the model can become without incohesion; a serious loss of cohesion would result from abstracting further to, say, the fact that either q1 q2 =r 2 equals 30 dynes or q1 q2 =r2 equals 15 dynes and the bodies also exert 15 dynes of some other kind of mutually attractive force. (Presumably, to introduce another species of non-negligible force would require crossing a gap in ‘causal similarity space’; the two disjuncts involve two quite different kinds of causal processes.) In contrast,

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small adjustments to q₁, q₂, and r that keep q1 q2 =r2 equal to 30 dynes is as contiguous in causal similarity space as small adjustments to the cannonball’s m and v that keep its momentum exceeding 20 kg m/s. The point is that presumably, the force’s magnitude is not in fact explained by the value of q1 q2 =r 2 . Rather, it is explained by the values of q₁, q₂, and r, along with Coulomb’s law and the absence of further influences. Strevens’s requirement that an explanation abstract as far as possible (without undue sacrifice of cohesion) effaces the distinction between property combinations that are explanatorily significant in their own right (such as a system’s having a given center of mass) and combinations that are mere conglomerates of the properties actually doing the explaining. As another example, take one of Fick’s laws of diffusion, namely, that the volume (V) of gas flowing across a tissue or membrane per unit time is directly proportional to the area (A) of the sheet, the difference ðP1  P2 Þ in the gas’s partial pressures across the sheet, and the gas’s solubility (S), and inversely proportional to the sheet’s thickness (T ) and to the square root of the gas’s molecular weight (m): V / AðP1  P2 ÞS=T√m Now ðP1  P2 Þ may well function as a unit in this explanation; pressure differences, not absolute values, are responsible for propelling the gas. But it is more doubtful that the entire right side of the above expression functions as a unit in an explanation. Yet as far as I can tell, maximal abstraction would drive us to this conclusion. There is a difference between a hodgepodge, ramshackle quantity such as reduced mass or AðP1  P2 ÞS=T√m, on the one hand, and momentum, density, work, or kinetic energy, on the other hand.

4. Dimensionless Quantities as Explanatorily Powerful Reducible Properties Various dimensionless quantities are well-known examples of reducible properties that explain. Many are designated by the names of the scientists who discovered their explanatory significance, such as the Reynolds number, Prandtl number, Biot number, and Grashof number. For example, for a fluid flowing down a pipe where ρ ¼ fluid’s density v ¼ fluid’s mean speed of flow d ¼ pipe’s linear dimension (e.g., its diameter, for a pipe with circular crosssection), and η ¼ fluid’s viscosity, the fluid’s having a ‘Reynolds number’ (Re) of n is identical to its having the above properties standing in a certain relation, namely, n ¼ ρvd=η. A fluid’s Reynolds number is plainly a reducible property. Nevertheless, it is commonly said to ‘govern’

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or to ‘control’ many features of the flow, such as whether it is laminar or turbulent— that is, whether the streamlines are parallel to the pipe’s long axis and hold constant, or whether they change drastically and irregularly as eddies and vortices form and dissipate. There is no simple function from the Reynolds number to the flow’s character. But two fluid systems that are alike in their background conditions (e.g., the pipe’s geometry and smoothness, the external disturbances it feels) are alike in whether the flow is laminar or turbulent if they have the same Reynolds number. The Reynolds number is not merely correlated with the flow’s character; it helps to explain the flow’s character: ‘In addition to geometry and pressure gradient, several other parameters influence separation. These include the Reynolds number as a very important parameter, with the wall’s roughness . . . and the wall temperature having less but occasionally significant influence.’³¹ Indeed, in fluid mechanics it is commonly said that the flow’s character is explained by Re (along with various background conditions) rather than by ρ, v, d, or η taken individually—in other words, is explained by the more basic quantities only through their relations to one another in forming Re. Reynolds himself expressed his discovery in these terms, and modern textbooks agree: [W]hat appears to be the dependence of the character of the motion on the absolute size of the tube, and on the absolute velocity of the immersed liquid, must in reality be a dependence on the size of the tube as compared with the size of some other object, and on the velocity of the body as compared with some ‘other velocity.’³² Reynolds realized that the transition from laminar to turbulent flow did not depend on the pipe diameter, the flow velocity, or the fluid viscosity individually, but on the nondimensional grouping of those three parameters that we now call the Reynolds number.³³

Dimensionless quantities such as Re are especially useful in the construction of physical models for predicting the behavior of systems to which it is difficult in practice to apply the relevant fundamental equations. For instance, suppose we want to predict the force F that a large industrial stirrer of length d will experience while stirring (with speed v) a fluid of density ρ. By law, F ¼ ρv2 d2 ΦðReÞ for some function Φ. Since in practice Φ cannot be predicted, the best way to predict F is to construct a smaller scale model of the stirrer with the same shape (and background conditions) so that Φ remains the same. If the model’s d is one tenth of the industrial stirrer’s, for example, but the model’s v is ten times the industrial stirrer’s, then (if the same fluid is used) they have the same Re. Then by using the model, the industrial stirrer’s F can be discovered empirically. So much for prediction; let’s turn to explanation. Why does the model work—why do it and the industrial stirrer experience the same force? An explanation is that they have the same ρ, the model’s v is ten times the industrial stirrer’s, the model’s d is one tenth the industrial stirrer’s, they have the same Re, and by law F ¼ ρv2 d2 ΦðReÞ for ³¹ Potter et al. (2012, 351).

³² Reynolds (1901, 53–4).

³³ Post (2011, 454).

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some unspecified function Φ. Notice that Re is an explainer on a par with ρ, v, and d despite being a reducible property. In contrast, though the law could be expressed in terms of two reducible properties—ρv²d² and Re—the former property is insufficiently natural to have explanatory power. I aim to account for this distinction. Here is another way to appreciate it. Just as two seesaws both tip to the right because they are alike in having ðmR xR  mL xL Þ> 0, so it is no coincidence that the modeled stirrer and the model experience the same force. They do so because they share certain other properties, notably Re. Likewise, it is no coincidence that two instances of fluid flow are both turbulent; they are alike in that respect because they are alike in their pipe geometry (and other background conditions) and have the same Re. Thus, having the same Re must be a respect of similarity, so Re must be natural enough to explain. An account of the difference between natural and unnatural reducible properties should account for the naturalness of all dimensionless quantities that function in the same way as Re.

5. My Proposal In each of the cases we have examined, a law L involving some reducible property P, together with some initial conditions that include P’s instantiation, entail some outcome. When is P natural enough for its instantiation to be eligible to join with L and other initial conditions in explaining the outcome? I propose that P is natural enough to explain if and only if there is an explanation of L in which P (that is to say, the combination of properties to which it reduces) enters as a unit. For example, L must have an explanation (which need not be its only explanation) in which it is not the case that some of the properties to which P reduces enter under separate auspices from others so that it is only somewhere in the course of the explanation of L that these various properties happen to find themselves grouped together so as to form P. Intuitively, P is natural enough exactly when P figures in L as a result of entering an explanation of L as a unit rather than as a result of precipitating out over the course of any explanation of L by a kind of algebraic ‘miracle.’ Let’s apply this proposal to our seesaw example. The seesaw’s mR xR  mL xL is sufficiently natural to join the tipping law in explaining the seesaw’s tipping to the right because it appears in the expression for the seesaw’s center of mass, which figures in the tipping law, and the tipping law (in turn) has an explanation in which the system’s center of mass (and hence its mR xR  mL xL ) enters as a unit rather than emerging from factors that have entered separately. One such explanation proceeds from energy conservation. The only external forces on a system in the tipping law’s scope are gravity and the force exerted by the support. Since the system is attached to its support, the supporting force cannot act through a distance and so only gravity can perform work on the system. Any change in the system’s gravitational potential energy as a result of gravity’s doing work on the system must be balanced by an equal and opposite change in the system’s kinetic energy. The gravitational potential

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energy of a system in the tipping law’s scope (i.e., under uniform gravity) is equal to the gravitational potential energy U of a hypothetical point particle having the system’s mass M and located at its center of mass: U ¼ Mgh, where h is the height of the center of mass. Here, then, is where the center of mass enters as a unit. A system initially at rest cannot begin to move, gaining kinetic energy, unless it thereby loses potential energy, and so if it begins to move, it must move so as to lower h. Moreover, it can begin to move exactly when there is a net force or a net torque on it, i.e., exactly when @U=@x 6¼ 0 in some direction x in which it is free to move. This condition is met if and only if its center of mass is not located directly above or below a point of support. (For example, if its center of mass is directly above a point of support, then if the system moves a little while remaining attached rigidly to its support, its center of mass does not change its height, so U remains unchanged: @U=@x ¼ 0.) In this way, the tipping law is explained—and the center of mass’s location (for the seesaw: ðmR xR  mL xL Þ=MÞ enters this explanation as a unit. (In so doing, the seesaw’s mR xR and mL xL also enter as units and so are also natural enough to help explain.) Of course, there are other ways to explain the tipping law in which the center of mass arises fortuitously in that the various, more basic properties figuring in the expression for the center of mass’s location enter by playing different roles and come together to form that expression only in the course of the tipping law’s derivation. For example, we could explain the tipping law by first finding the net torque τ ¼ ∑ðri  Fi Þ produced by gravity to be τ ¼ ∑ðri  mi gÞ and then massaging this sum to isolate an expression for the center of mass’s location. (From there, the explanation would take the internal torques’ sum to vanish, by Newton’s third law, and would then set the total torque equal to Iα.) The various properties to which the center of mass’s location reduces thus enter this explanation under separate auspices: the ri’s as the lever arms in the component torques, the mi’s as factors in the forces creating each of those component torques. They do not all enter together, arriving pre-packaged in the expression for the center of mass’s location—as they do in the above explanation from energy conservation.³⁴ On my proposal, the existence of a single explanation of the tipping law in which the center of mass’s location enters as a unit suffices to make that property natural enough to explain.³⁵ ³⁴ Perhaps there is another explanation of the tipping law from energy conservation that does not begin from an expression ðU ¼ MghÞ for the system’s gravitational potential energy in which the center of mass’s location figures as a unit. Perhaps there is an explanation that begins with the gravitational potential energies of each point particle and then massages the sum of those energies until Mgh emerges. It is not obvious to me that this argument is in fact explanatory—that the expression for the potential energy of a configuration derives from the expression for the potential energy of a point particle. The order of explanatory priority might be the other way around, or the two laws may be equally basic. But even if the above argument does explain, it does not undermine the explanatory power of the tipping law’s derivation from energy conservation and U ¼ Mgh. ³⁵ One does not need to know all of the details of such an explanation of the tipping law in order to have good reason to believe that there is such an explanation, and hence that the seesaw’s ðmR xR  mL xL Þ is natural enough to explain. Likewise, one who has not examined all possible explanations of the tipping law might justly believe that there is no explanation into which ðxR  mL xL =mR Þ enters as a unit, on the

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It is not uncommon for some reducible property, though emerging ‘miraculously’ in the course of a causal-mechanical explanation of some law, to enter instead as a unit when that law is explained by conservation laws and to qualify thereby as natural enough to explain.³⁶ For an especially dramatic example, consider the weight of the fluid displaced by a submerged body as a reducible property explaining the buoyant force on the body. The covering law in this explanation is Archimedes’ Principle: that the buoyant force on a body surrounded by an ideally incompressible, non-viscous fluid in a container at rest in a uniform downward gravitational field equals the weight of the fluid displaced by the body. Let’s contrast two explanations of this principle.³⁷ It is explained hydrostatically by the difference in pressure on the upper and lower surfaces of the submerged body. The weight of the displaced fluid does not enter this explanation as a unit. Rather, in the course of the derivation, there arise separately various factors that eventually come together to form an expression that turns out equal to the weight of the displaced fluid: Aρgh (for fluid density ρ, taking the submerged body for simplicity as a cylinder of height h with horizontal top and bottom surfaces, each of area A; any body can be taken as consisting of cylindrical elements). The various factors in this expression enter the derivation separately: ρg as the rate at which the pressure changes with height, h as obviously the difference in height between the body’s upper and lower surfaces, and A as the factor by which the pressure on an upper or lower surface must be multiplied to yield the force downward or upward on that surface. The principle can also be explained by energy conservation, according to which the work done by the force just sufficient to raise the submerged body by some arbitrary distance d is equal to the energy thereby added to the system. With the body’s rise, its gravitational potential energy rises by Wbd, where Wb is the body’s weight. As the body rises, fluid flows downward to fill the space it vacates. As a result of its journey, the body effectively trades places with a fluid parcel of the body’s size and shape that was initially located at a distance d above the body’s initial position. That parcel thus descends by d, so its gravitational potential energy diminishes by Wpd, where Wp is the parcel’s weight. Setting the work done equal to the net increase in the system’s

grounds that this combination of quantities seems quite arbitrary. (Such a person might say, ‘If even this arbitrary quantity enters as a unit into some explanation of the tipping law, then presumably for any combination at all of those four quantities, there is some explanation into which it enters as a unit— which surely cannot be.’) Of course, further investigation might reveal that an arbitrary-looking quantity is not actually arbitrary after all. Nevertheless, one who has examined some but not all possible explanations of the tipping law, and also has some background knowledge of physics, may be warranted in being very confident that a given arbitrary-looking quantity fails to enter as a unit into any explanation of the tipping law. ³⁶ Ohanian (1991, 182), for example, emphasizes that a law like the tipping law has not only a causalmechanical explanation, but also an explanation from energy conservation. ³⁷ For more detailed presentations of these explanations, see Lange (2011).

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potential energy, we find Fd ¼ Wb d  Wp d. Therefore, F ¼ Wb  Wp . By contrast, had there been no fluid, then the force just sufficient to raise the body would have been simply Wb—just enough to balance the body’s weight. But with the fluid present, we need less force by Wp: the weight of the fluid displaced by the body. That upward force supplementing F is the buoyant force. In this explanation, Wp enters as a unit—in contrast to the hydrodynamic explanation. This explanation, then, explains why the buoyant force turns out to equal the weight of the displaced fluid without portraying this fact as arising by algebraic coincidence. In other words, the conservation law explanation reveals that each of the factors to which the weight of the displaced fluid reduces appears for the very same reason in the resulting expression for the buoyant force; they appear there together because they enter the derivation together rather than from different places. This explanation thus makes Wp natural enough to join Archimedes’ Principle in explaining the magnitude of the buoyant force on a given body.³⁸ Even if the various properties to which a given reducible property reduces enter the explanation together, it does not follow that the reducible property enters as a unit. Those properties may all enter together, but not in an expression of the relation among them that is identical to the reducible property. For instance, take the explanation of the tipping law that I gave earlier in which the location of the center of mass (for the seesaw: ðmR xR  mL xL Þ=MÞ enters as a unit. Suppose we add a step to the end of this explanation: having derived the law that the seesaw tips to the right if ðmR xR  mL xL Þ> 0, we draw the conclusion that the seesaw tips to the right if ðxR  mL xL =mR Þ> 0, thereby explaining why this derivative law holds. Now the various properties to which the seesaw’s ðxR  mL xL =mR Þ reduces all enter the derivation together. But they do not enter as the unit ðxR  mL xL =mR Þ; rather, they enter in ðmR xR  mL xL Þ in the expression for the seesaw’s center of mass. Thus, the reducible property ðxR  mL xL =mR Þ does not enter as a unit, and so on my proposal, this explanation does not make the seesaw’s ðxR  mL xL =mR Þ sufficiently natural to explain. Let’s look at another seesaw example: a seesaw on each arm of which is placed the same number N of bodies of negligible size having various masses, all piled up at the same distance x ¼ xR ¼ xL from the fulcrum. For this special case, consider the covering law that the seesaw tips to the right if the average mass of a body on the right is greater than the average mass of a body on the left. Plainly, though this derivative law predicts the outcome, it does not explain the outcome; the average masses on the right and left are not the explanatorily relevant features. The two averages

³⁸ Notice that it is not circular to say that Wp is natural enough to join Archimedes’ Principle in explaining the magnitude of the buoyant force on a given body by virtue of its entering as a unit in some explanation of Archimedes’ Principle. Firstly, the two explanations have different targets: the former explains the magnitude of the buoyant force on a given body, whereas the latter explains Archimedes’ Principle. Secondly, the explanation of Archimedes’ Principle does not itself depend on Wp’s naturalness.

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(mavR ¼ mR =NR and mavL ¼ mL =NL ) are merely artifacts of properties that can explain the outcome: the total masses mR and mL on the two wings (along with the seesaw’s other conditions, including xR ¼ xL ). (An explanation could just as well proceed from the masses of the individual bodies.) The seesaw tips to the right because there is more mass on the right (and various other conditions hold, crucially xR ¼ xL ), not because there are equal numbers of bodies on the two arms and the average mass of a body on the right is greater than the average mass of a body on the left. Talk of averages here is just an unnecessarily roundabout way of getting at the explanatorily relevant fact that there is more mass on the right. My proposal accounts for the averages being too unnatural to explain. Suppose we take the explanation of the tipping law that I gave earlier and, having explained the law that the seesaw tips to the right if ðmR xR  mL xL Þ> 0, we now add some further steps at the end to derive the law covering this special case: since xR ¼ xL , the seesaw tips to the right if ðmR  mL Þ> 0, which is to say (since N > 0) if ðmR  mL Þ=N> 0, i.e. (since N = NR ¼ NL), if ðmR =NR  mL =NL Þ> 0, i.e., if mavR > mavL . Plainly, mavR and mavL do not enter as units; mR (mL) enters under different auspices from NR (NL) and they coalesce into the average only at the end. Intuitively, whereas mR and mL enter naturally (back in the tipping law’s derivation), NR and NL enter artificially—merely to generate averages—and that is why the average masses fail to explain. My account captures this intuition. My proposal likewise deems reduced mass explanatorily idle. In my earlier examples, the explanandum in a two-body problem was entailed by premises including the pair’s reduced mass μ ¼ m1 m2 =ðm1 þ m2 Þ and the law that the first body moves relative to the second in the manner of a point body feeling the same force as the first body feels but with mass equal to the pair’s μ. On my proposal, whether μ is natural enough to explain depends on whether this law has an explanation in which μ enters as a unit. Here is a standard explanation of this law: By Newton’s second law, m1 a1 ¼ F12 ; m2 a2 ¼ F21 where Fij is the force on body i exerted by body j, mi is the ith body’s mass, and ai is the ith body’s total acceleration (since in a two-body problem, the only forces are their mutual interactions). By Newton’s third law, F12 ¼ F21 : Hence, m1 a1 ¼ m2 a2 ;

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so a2 ¼ ðm1 =m2 Þa1 : The first body’s acceleration relative to the second is then   a1  a2 ¼ a1 þ ðm1 =m2 Þa1 ¼ ð1 þ m1 =m2 Þa1 ¼ ðm1 þ m2 Þ=m2 a1   ¼ ðm1 þ m2 Þ=m1 m2 m1 a1 ¼ F12 =μ: Obviously, μ does not enter this explanation as a unit. Rather, m₁ and m₂ enter playing different roles and μ emerges fortuitously. Thus, on my view, μ and ðm1 þ m2 Þ and m₁m₂ are all too unnatural to explain here.³⁹ In Section 4, I suggested that an account of why certain reducible properties but not others are natural enough to possess explanatory power should account for the naturalness of various dimensionless quantities, such as the Reynolds number Re. On my account, these quantities acquire their explanatory power by virtue of entering as units into the ‘dimensional explanations’ of various derivative laws. In a dimensional explanation,⁴⁰ the explanans is that some property v stands in a ‘dimensionally homogeneous’ relation to some subset of other properties s, t, u . . . —that is, a relation that holds for any system of units (e.g., meters and feet) for the various dimensions (e.g., length) of the quantities so related. From this premise, various features of the law v ¼ f(s,t,u . . . ) can be deduced and thus explained as required by the dimensional architecture. If there is some dimensionless combination of s,t,u . . . , then the dimensional argument can explain the law only up to an unspecified function of that combination; dimensional analysis alone is unable to specify the function. That is how a dimensionless combination enters as a unit into the dimensional explanation and thus becomes natural enough to possess explanatory power. For instance, let’s look at a dimensional explanation of the law appearing in the example from Section 4: that the force F that a large industrial stirrer of length d will experience while stirring (with speed v) a fluid of density ρ is equal to ρv²d² Φ(Re) for some function Φ. The explanans is that F stands in a dimensionally homogeneous relation to d, ρ, η, and v. It follows that the law takes the form F ¼ da ρb ηc ve Ψðρvd=ηÞ for some unspecified function Ψ, since ρvd=η is dimensionless. Thus, the combination ρvd=η forming Re enters as a unit; it does not

³⁹ This conclusion presupposes that all explanations of the derivative law involving μ are like this one. If there were another explanation into which μ entered as a unit, then my account would offer a different verdict. I do not know of such an explanation; the law can be explained by a Lagrangian approach to the two-body problem, for instance, but once again, μ does not enter as a unit into the Lagrangian. ⁴⁰ According to Lange (2009a).

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gradually coalesce in the course of the derivation. The rest of the explanation involves giving the dimensions of each quantity and then solving for a, b, c, and e: F Mass 1 Length 1 Time 2

d ρ η v 0 1 1 0 1 3 1 1 0 0 1 1

so 1 ¼ b þ c so 1 ¼ a  3b  c  e so 2 ¼ c  e

Hence; e ¼ 2  c b¼1c a¼2c So F ¼ d2c ρ1c ηc v2c Ψðρvd=ηÞ ¼ ρv2 d2 ðρvd=ηÞc Ψðρvd=ηÞ ¼ ρv2 d2 Φðρvd=ηÞ for some unspecified function Φ. By entering into and then behaving in this explanation as a unit, Re acquires explanatory significance.⁴¹ Other dimensionless quantities likewise possess explanatory power in virtue of functioning as units in dimensional explanations. For example, from a dimensional explanation of the law P ¼ WΦðh=lÞ [for some unspecified function Φ] concerning the tension P of a length l of cable having weight W and sag h between two points at the same height and a fixed distance apart, Douglas concludes that ‘the ratio h/l is a determining factor in the relationship rather than the separate quantities h and l.’⁴² I do not interpret Douglas as claiming that h and l have no explanatory power themselves, merely that they do not explain except relative to each other—that is, except by way of h/l explaining.⁴³

6. Conclusion I have been proceeding as if the distinction between explanatorily powerful and powerless (or natural and unnatural) reducible physical properties were a sharp distinction. But this seems to me an oversimplification. The distinction appears to be more plausibly a matter of degree; some properties may be sufficiently natural to have some but insufficiently natural to have much explanatory power. My proposal leaves opportunities for such intermediate cases. For instance, a property P might enter a given explanation of L as a unit, but only because this explanation of L is not ⁴¹ The precise explanation of the onset of turbulence is a notorious open question. Scientists justly expect Re (as the ratio of inertial to viscous forces) to enter as a unit into an explanation of whatever derivative laws cover this phenomenon (see the above passage from Reynolds). But as of this writing, no general explanation has been found. ⁴² Douglas (1969, 37). ⁴³ Suppose we take seriously the idea that the property of having a given value of the Reynolds number qualifies as a natural property (i.e., a genuine respect of similarity) by virtue of its role in dimensional explanations, which are not causal explanations (i.e., do not derive their power to explain from supplying information about the world’s network of causal relations). The view (Fodor 1988; Shoemaker 1984, 206–33) that physical properties qualify as natural by virtue of their causal roles would then have to be amended to ‘becausal’ roles.

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much of an explanation; it derives L so trivially from facts standing so close to L that it supplies little information about where L ultimately comes from. If this is the only explanation of L that P enters as a unit, then P has little naturalness but it is not an utterly arbitrary combination of quantities. Let’s look briefly at a possible example. In Section 3, we found that according to Strevens’s account of scientific explanation, the force between a pair of point charges suddenly released at rest in an otherwise empty universe (feeling negligible nonelectrostatic forces) is explained by Coulomb’s law (concerning the electric force between two point charges that have long been at rest) together with the pair’s q1 q2 =r 2 . I suggested that in fact, this reducible property is too unnatural to explain. But perhaps this quantity does not quite fall at the extreme ‘unnatural’ end of the spectrum. On my proposal, this property is natural enough to explain exactly when it appears as a unit in an explanation of Coulomb’s law. Does it? According to some authors,⁴⁴ Coulomb’s law is fundamental; that is, it has no explanation. In that case, trivially q1 q2 =r2 does not enter as a unit into an explanation of Coulomb’s law. Other authors regard Coulomb’s law as explained by being derived from Maxwell’s equations.⁴⁵ In that case, my account again deems q1 q2 =r 2 to be explanatorily powerless since q1 q2 =r 2 does not enter this explanation as a unit. However, Coulomb’s law is a special case of the general formula⁴⁶ for the electric force Fe between two point bodies with unchanging charges (q₁ and q₂) that have been in arbitrary relative motion: 0

00

Fe ¼ q1 q2 ½u=R2 þ ðR=cÞðu=R2 Þ þ ðu =c2 Þ where R is the distance between the body feeling the force and the position of the body exerting the force at the ‘retarded time’ (that is, the moment when an electromagnetic wave (traveling at speed c) arriving at the body feeling the force would have left the body exerting the force), a prime (΄ ) denotes the rate of change of the preceding quantity, a double-prime (΄΄ ) denotes the rate of change of the rate of change of the preceding quantity, and u is the unit vector pointing from the body feeling the force directly away from the other body’s position at the retarded time. Coulomb’s law follows trivially from this formula for the case where the bodies have not been moving: all rates of change are zero (so the second and third terms vanish) and R in the static case is equal to the bodies’ current separation. If this quick derivation is an explanation of Coulomb’s law, then it is a rather slight one; Coulomb’s law just falls trivially out of the more general formula. So although q1 q2 =r 2 enters this derivation as a unit (in the formula’s first term), it does not thereby merit a place far along toward the explanatorily powerful end of the spectrum.⁴⁷

⁴⁴ See, e.g., Serway and Jewett (2008, 642). ⁴⁵ For the derivation, see Feynman et al. (1963, 21); Janah et al. (1988). ⁴⁶ This formula is given by Feynman et al. (1963, 21), who fail to mention that it was first published by Heaviside in 1910 (Heaviside 1912, 438). ⁴⁷ There are other ways on my proposal for intermediate cases to arise. Suppose that property P fails to enter as a unit into any explanation of any law that, together with some initial conditions that include

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To conclude: The proposal that I have offered seems a promising way to account for why certain reducible physical properties but not others are natural and explanatorily powerful. It gives the correct verdicts regarding various examples I introduced earlier, including the seesaw’s center of mass ðmR xR  mL xL Þ and xR  mL xL =mR , its mavR and mavL, a two-body problem’s reduced mass, a submerged body’s Wp, and a fluid’s Reynolds number. This approach captures the intuition that a reducible property fails to explain when it is an arbitrary conglomerate of properties. A reducible property that enters the covering law’s explanation as a pre-packaged unit, rather than emerging fortuitously out of a welter of algebra, is no arbitrary combination.⁴⁸

P’s instantiation, entail a given outcome O. However, suppose that P nevertheless enters some explanation of L as a unit, where L is a law that performs considerable explanatory work (though not in explaining O). Then P’s role in L’s explanation might give P some degree of naturalness—enough for O’s entailment from P’s instantiation (together with other facts) to possess some power to explain O. ⁴⁸ I am grateful to audiences at the University of Miami, the University of Cologne, and the Metaphysics of Science Summer School at the University of Helsinki for their helpful feedback regarding this chapter.

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11 Laws and Their Exceptions Stephen Mumford

1. Introduction Laws and exceptions seem to be in conflict. A law is supposed to be universal, applying at all times and places, whereas an exception must be a contravention of such universality. It might seem that if there is an exception, then there is no universality and we cannot, for that reason, have a law in the first place. But might there be some room nevertheless for the notion of an exception to an otherwise general rule? After all, the very meaning of exception seems parasitic upon the notion of a law, as reflected in the phrase ‘the exception that proves the rule.’ An event or action counts as an exception only if it has some association with a law that it somehow breaks. That association is clearly a difficult one to describe because, while being connected with a law or general rule in some way, an exception is something that escapes it. The aim of this chapter is to offer a syncretic solution showing how there could be exceptions to natural laws or laws of nature. It is worth noting for a start that other notions of law, such as non-natural laws, do accommodate the exceptional. There could be a law of the land—the legal case of law—that permitted exceptions. A law might state that everyone is obliged to make their tax return by April 1, but then be followed by a list of exceptions, such as for those serving in the armed forces overseas. Someone might request that the authorities make an exception for them. Moral laws might also have exceptions. Although we think that we should not end another person’s life, some people allow that we should make an exception in specific circumstances; for instance, for someone terminally ill with a progressive, degenerative disease, who is incapable of taking their own life and requests another to do it on their behalf. One motivation for moral particularism is that there is no moral law that holds in every single situation.¹ Laws of nature seem to have special difficulties in allowing exceptions, however. Natural laws are supposedly outside the human sphere of influence so we cannot simply choose which ones we want, nor to be exempt from some of them. They are ¹ Dancy (2004).

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descriptive of nature’s workings rather than prescriptive of our own desired outcomes. In the legal case, in contrast, we can make a decision to exclude certain categories of person or circumstance from the scope of the lawful prescription. We might think that the moral case is like the legal one in that we are attracted to adopt moral systems that are sensitive to the need for exceptions to be made. Moral realists might think that moral laws are more like laws of nature, in being outside our influence and control, and this is a controversy into which I shall not enter. But in any case, an objective morality could still be one that permitted exceptions. Laws of nature, in contrast, are descriptive of the regularities in nature and this suggests that prima facie if something is less than regular—because it has exceptions—then it is not a law. A moral law, in contrast, remains prescriptive rather than descriptive even if that prescription has an objective standing. It should be noted, on the other side, that there is a view that makes laws of nature look to be within the human sphere and that human practices—especially our epistemic ones—make the laws of nature. Perhaps this could be a way to interpret Cartwright’s notion of a nomological machine,² where we sometimes set up the experiments that produce laws of nature. More obvious theories that support such a view, though, are epistemic senses of law that concern our expectations and predictive practices.³ On such a view, what makes something a law is merely the central position it holds in our theory. I am going to set such views aside, however, because the particular problem I want to address is how, if at all, the notion of an exception can be squared with that of a real, objective, and mind-independent notion of law of nature. The difficulty raised by such a case is that alluded to at the very beginning. To have a law, we must, it seems, have something with universal scope over a certain domain. But an exception is intended to be precisely a violation of such universality. There is, therefore, a prima facie contradiction in the very notion of an exception to a law. To be an exception, something (an event, a state of affairs, for example) must in some way contravene a law. But if it does so, the putative law that is contravened is thereby not of universal scope. That would mean, by many lights, that it is not a law. And, in turn, it would mean that our original putative exception to the law is not, after all, an exception to a law. Instead, it might be alleged, we were simply wrong in the first place about the law in question. Perhaps we did not quite have the right formulation and the putative exception indicates instead that we need a more accurate statement: one which incorporates the recalcitrant case. We might then think that the truly and accurately formulated real laws of nature will be entirely exceptionless. If we take this line, therefore, the notions of law and exception are indeed in conflict. Laws of nature do not have exceptions. The problematic here is similar to the one that motivated Hume to reject the possibility of miracles.⁴

² Cartwright (1999, ch. 3).

³ For example, Ayer (1963).

⁴ Hume (1748/2007, sec. x).

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In this chapter, however, I attempt, to an extent, to reconcile the notions of law and exception. I do this by defending a certain conception of laws, and nature in general, which might be called neo-Thomist. This philosophy of nature is one that I think can be considered on its individual merits in abstraction from its original theological context in Aquinas’s work. While inspiration for the theory may come primarily from Geach’s account of Aquinas,⁵ we find similar ideas in John Stuart Mill’s and C.S. Peirce’s views of laws,⁶ as well as newer work on dispositional theories of causation,⁷ that all aim to be entirely naturalistic accounts. The reconciliation also makes use of some recent ideas promoted by Lipton and Lowe.⁸ The solution is that while laws are in a sense universal in scope, their content concerns what is disposed to happen only—what tends to be—rather than what is necessitated. What we take to be exceptions will typically be cases where those dispositions are, for whatever reason, unmanifested. The exceptions are not, therefore, to the laws themselves, but instead point to the relatively commonplace and unproblematic notion that dispositional processes can fail to manifest. Given that the law has only a dispositional import, however, it is not violated, and thus remains a law applying to everything within its scope, even if there are exception cases. A common attempt to accommodate exceptions into laws is to say that such laws are true ceteris paribus, or all else being equal. This strategy is notoriously problematic as it seems difficult to provide non-trivially true content for the ceteris paribus (henceforth cp-) clause. But here I will offer an interpretation of the cp-clause that I take to be neither trivial nor false, namely one that is taken as indicative that the law statement has to be understood as having dispositional force only. I will thus be trying to explain the relationship between laws and their exceptions via a non-trivial interpretation of the use of ceteris paribus.

2. The ceteris paribus Problem Understanding the operations of nature in terms of the operation of laws of nature is a relatively recent trend in the history of human thought. Bacon⁹ was an early thinker to do so, applying his background training as a lawyer to advance a legal model of the world, with God, of course, as the lawmaker.¹⁰ Newton then gave this philosophical idea some scientific credentials when he presented his three laws of motion and one of gravitational attraction.¹¹ Since Hume, philosophers have conceptualized these laws as being exceptionless regularities in nature. This contrasts with the Thomistic view, as we shall come to see. Once modern logic was developed, it became natural for empiricist philosophers to articulate law statements in the form of universally quantified conditionals, such as 8x (Fx ! Gx). The form was proposed to cover a ⁵ Geach (1961). ⁶ Mill (1843/1973) and Peirce (1892/1991). ⁷ Mumford and Anjum (2010, 2011a, 2012). ⁸ Lipton (1999) and Lowe (2006). ⁹ Bacon (1620/1985). ¹⁰ Gower (1997, 52). ¹¹ Newton (1687/1934, axioms).

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wide range of laws, from the macroscopic and mundane, such as that all ravens are black, to the microscopic laws of physics and scientifically respectable laws such as Boyle’s law of the inversely proportional variation of volume and pressure on a gas, Coulomb’s law concerning electrostatic attraction, and the gravitation law. It is not clear that anyone explicitly thought that this was all that there is to a law of nature but Humeans continue to be attracted to the view of laws as regularities.¹² A problem, however, is that such laws, even those that are scientifically respectable, at least prima facie seem subject to exceptions. There are albino ravens, for example, which are still ravens even though they are not black. Boyle’s law applies only to ideal gases which are held at stable temperatures. Even then, the law will not be borne out in reality if there are too many collisions among the particles in the gas. If one is to maintain that the correct logical form of a law statement is a universally quantified conditional, then such exception cases present a problem. Even in physics, regularities are often interrupted by the presence of other factors. A possible option is to discount the exception. One could try to say that the albino bird is not, after all, a raven. The law could be maintained that everything that is F is G by saying that this non-G is, after all, also a non-F. But this response is implausible, looking like an ad hoc evasion of the challenge. There are, no doubt, many good reasons to think of the albino bird as a raven. It was born of a raven, it has a very close gene structure to that of ravens, it lives and mates with ravens, and so on. And it seems implausible automatically to rule out exceptions by saying that they are nonFs, which is to suggest that All Fs are Gs is, after all, analytic. Almost no defender of laws of nature has wanted that conclusion. Candidate laws should be, at least in principle, falsifiable, so the issue of whether there is a non-G F should be at the very least synthetic. Suppose, then, that there are no de facto perfect regularities. Less-than-perfect regularities in nature might nevertheless hold for the most part, with only a few exceptions, and might prove fairly robust under testing and useful in prediction. A response is to accept that these really are the laws but that they should be cp-qualified. The task then is to explain what this qualification means. What would be a cp-law? If we append the cp-clause on to the standard form of a law, we get: (CP1):

8x (Fx ! Gx), cp.

This is immediately problematic, however. The new formulation gives universality with one hand, because it is universally quantified, but then takes it away with the other hand, because it is cp-qualified. There is a prima facie appearance of self-contradiction. CP1 might even be oxymoronic. That appearance could be only superficial, however. Perhaps there is some way of filling in the content of the cp-clause such that it is not self-contradictory.

¹² For example, Swartz (1995). For a more sophisticated regularity view, see Psillos (2002).

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Here there are again problems, however. Suppose one tried to offer some positive content in place of the cp-clause. What if one said it meant that all Fs were Gs unless they were also Hs, for instance. With this suggested meaning of cp, we would have: (CP2):

8x ((Fx ! Gx), unless Hx).

Perhaps the true law should be understood as that all ravens are black unless they are albinos. The obvious problem with CP2 is that there could be some further reason why something was not black even though it was a raven, such as that it was affected by radiation. We could add this as a further clause: all ravens are black unless they are albinos or irradiated. But as long as this cp-clause is cashed out in terms of a finite list of exceptions, there will be at least the possibility of some further case that is not included in the list: the raven is a mutant, has been dyed, and so on. If we have a finite list, then the possibility of an exception, that renders the law statement false, remains a possibility. Thus only token progress will have been made in dealing with exceptions by a cp-clause. Another option, of course, is to try to rule out all such exceptions with some kind of catch-all clause, which will automatically take any possible exception within its scope. This will not be a list of types of exception at all, but more an attempt to refer to the exceptions as a whole, in some way ϕ. That would then give us: (CP3):

8x ((Fx ! Gx), unless ϕ).

The danger of this approach is triviality. We need to know the specific content of the ϕ-clause, of course; but the problem is its function as a catch-all. We have already said that it is intended somehow to include every possible exception. But that is effectively to say that all Fs are Gs unless they are exceptions. If there is not some independent way of saying what it is to be an exception, then this amounts to saying that all Fs are Gs unless they are not, which is clearly trivial.

3. Saving cp-Clauses from Vacuity There are some recent attempts to save cp-laws from vacuity. Pietroski and Rey, for example, claim that cp-clauses ‘are “cheques” written on the banks of independent theories.’¹³ This means that if there are exceptions to the law in question, they can in principle ultimately be explained in terms of other laws. The cp-clause means that there can be an exception to a law but only because of interfering factors perfectly explicable in other nomological ways. This, they say, renders the cp-clause non-trivial because there is the possibility that no such independent explanation exists for a particular exception. Were that so, we would know the candidate law to be false. ¹³ Pietroski and Rey (1995, 81).

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I will go on in Section 6 to endorse the spirit, though not the detail (which I omit here), of Pietroski and Rey’s account. But I do so in a way that less validates the notion of an exception and emphasizes more the idea of each law describing a dispositional contribution to events. One reason a law has an apparent exception is because what actually happens is determined by many laws working together. Laws can thus cut across one another and they might do so ubiquitously. By this I mean that every single law could have an exception if, for every law, there is another that counteracts it. This need not require a fundamental, exceptionless law, at the bottom of them all, that interferes with other laws but has nothing interfering with it. Instead, law A could be counteracted because of law B, law B could be counteracted because of C, C by D, and then D by A. There are circumstances in which every law could be less than universal, therefore. If one insists on interpreting laws along the lines of universal regularities, it seems at least a contingent possibility that there are none. Every pattern in nature might be less than universal due to interference by some other factor. What is acceptable about Pietroski and Rey’s account, therefore, is the idea that the exceptions can have explanations elsewhere. But I will nevertheless be going on to give a dispositional reading of laws in which it is questionable whether anything counts as a genuine exception to a genuine law. Schrenk is sensitive to this issue and differentiates real from pseudo exceptions.¹⁴ A real exception cannot be merely epistemic. It cannot be just a result of our partial consideration of a situation or partial knowledge of a law. The law might be too roughly and vaguely stated, for example, and may admit exceptions as a consequence. But this of course does not mean that the correctly stated precise law, including further specifications, also has exceptions. Although recognizing that putative exceptions can be spurious, Schrenk nevertheless endeavours to defend two genuine cases of real exceptions to laws. The first kind of case is where a fundamental law has an exception at a particular space-time region.¹⁵ This region is just like any other and thus the law cannot be refined further so as to exclude those properties found at such a region. Rather, this region just happens to be one in which the law does not apply for no further reason. The second kind of case is what Schrenk calls singularities. These are theoretical instances in physics in which, under certain circumstances, all the laws of science break down.¹⁶ One kind of singularity is a black hole in which, according to some physical theory, chaos reigns.¹⁷ Schrenk’s first case violates one rule that is conceptually central to the notion of a law: that it names no individual. In this case, the individual so named is a particular space-time region. Schrenk accepts that he is doing this but defends it.¹⁸ The reason we want to exclude individuals from our laws is that naming them would allow us to formulate laws of a very narrow scope, restricted to some very local regularity, ¹⁴ Schrenk (2007, 25–6). ¹⁷ Schrenk (2007, 58).

¹⁵ Schrenk (2007, 45). ¹⁸ Schrenk (2007, 48).

¹⁶ Schrenk (2007, 54).

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such as in the case of all the coins in my pocket being silver. But Schrenk defends his own case, noting that it still leaves us laws of a very wide scope: that apply everywhere else except this very small local region. By referring to, for purpose of exclusion, this small region of space-time, we are salvaging an otherwise useful law. We can accept Schrenk’s claim that the two cases are different, but are they enough for us to relax, or even jettison, something so central to the notion of a law as its generality? Apart from the question of whether individuals can be named in our laws, it seems that the acceptance of real exceptions is a rejection of this generality; but it is not clear that it is ever worth rejecting. The second kind of case—the singularity—raises the same question but another in addition. The case is so theoretical and so little understood that at best it provides only tenuous grounds for the rejection of something so conceptually central as universality in our notion of a law. Schrenk’s aim is not, however, to prove that such exceptions to laws are actual. His purpose is only to show that the notion of a law is consistent with it also having a real exception. It doesn’t matter, therefore, whether there is any empirical evidence for the existence of so-called singularities but only that they are possible. And perhaps merely that they are conceptually possible is enough to show that there can be cp-laws, whether or not the theoretical physics turns out to be right. But has it really been demonstrated that exceptions to genuine laws are conceptually possible in this way? The alleged exceptions involve the violation of a central component of our notion of a law of nature. For that reason there has to be at least some doubt that we have both a law, L, and a genuine exception to it. Suppose, for instance, that there was not just one spatio-temporal region in which L did not hold. Suppose there were many and, as in the original case, there is nothing special about any of the regions in which L does not hold. Wouldn’t we then start to question whether L really is a law rather than just some non-nomic contingency that is fairly widespread but less than universal? And in the case of black holes and other singularities, would we really have to allow them as genuine exceptions? Couldn’t the laws be specified in such a way that they apply only in standard, non-singularity cases? Laws apply universally, but only over a specific domain. Laws state that ravens are black and that electrons have negative unit charge, not that everything in the universe has these properties. It is only the things that are F that are G, for example. It is not clear that singularities such as black holes and white holes should be included within the domain of any known laws of nature and, thus, within their true domain such laws might be exceptionless. All the putative exceptions may remain merely epistemic. It might be possible to answer some of these questions satisfactorily. But I will be going on to develop a view of laws in which we do not need to resolve these matters and challenge the conceptual boundaries of natural laws. I will be setting out a view in which exceptions are given a proper place but that they concern the manifestations of laws, not the laws themselves. Such laws can remain universal in scope as long as they

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are not about actual occurrences and thus not claims of de facto regularities. Laws are about what tends to be, and thus remain true even though not every instance manifests the tendency.

4. A Neo-Thomist Philosophy of Nature Schrenk’s strategy is to challenge the view that laws of nature must be exceptionless regularities.¹⁹ I also challenge this view, but for different reasons. Schrenk thinks laws needn’t be exceptionless regularities. I think, instead, that laws are not about exceptionless regularities. Schrenk defends a kind of qualified regularity, where the possible exceptions come in two kinds: spatio-temporal regions and singularities. But I argue instead that the laws should be understood dispositionally, following a view of the world that goes back at least as far as Aquinas.²⁰ The question of exceptions then becomes slightly more complex. In one sense, there will be many exceptions but only in that there are cases where the dispositions described in the law statement fail to manifest themselves. But as Schrenk himself says, these should not count as genuine exceptions at all: they are only pseudo-exceptions.²¹ Laws understood dispositionally can then be understood as having no genuine exceptions at all. In the dispositional account of laws the problem of exceptions is thus dissolved. The account is neo-Thomist in that it takes laws to be about dispositional or tendential relationships in nature only, rather than universal regularities. This may seem to be positing only a very weak kind of relationship, one that doesn’t give us as much as the universal feature of laws. But this is not the case. A disposition or tendency can be universal in that it applies to every particular within a domain. The manifestation of that tendency may be less than universal but that is another matter. A disposition provides something that is more than pure contingency. Out of all the many possibilities, a disposition is for only one kind of manifestation, even if that manifestation may get thwarted by other factors.²² The dispositionalist thinks this a more accurate account of nature than laws as exceptionless regularities. Being an F only disposes towards G, it does not necessitate it. Central to the notion of dispositionality is the possibility of prevention of manifestation. A fragile object only disposes towards breaking when dropped. It might land in such a way that it does not break; or some other factor might at the same time dispose away from breaking. As Schrenk argues,²³ and as Hume had before him,²⁴ there is always the possibility of interference. It might even be suggested that unless prevention is possible, we are not employing a disposition concept at all because any property that was necessarily manifested would be best thought of as a categorical property.²⁵ ¹⁹ Schrenk (2007, 35). ²⁰ As interpreted by Geach (1961). ²² Mumford and Anjum (2011a, ch. 8). ²³ Schrenk (2010). ²⁴ Hume (1739–40/1888, 161). ²⁵ Mumford (2006, 481).

²¹ Schrenk (2007, 25–6).

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What this means for the case of causation, and hence causal laws, is that a cause does not necessitate its effect, it only disposes towards it.²⁶ This is because causation is accounted for in terms of the manifestation of dispositions, offering a richer conception of cause than that in modern philosophy and since, which attempts to reduce causation down to the notion of efficient causation alone. For what it’s worth, an efficient cause would be akin to the stimulus of the disposition; but we need more than just stimuli to explain the occurrence of effects. Objects receiving the same stimuli may respond differently because they have different dispositions. The disposition may, thus, correspond closer to Aristotle’s notion of a formal cause. In cases of causation, then, some further natural process could always interfere and often does: what actually occurs will typically be a result of many dispositions acting together. Where P necessitates Q, then whenever we have P we will have Q. There can thus be an antecedent strengthening test of necessity: we should be able to add anything to P and still have Q. Supposing it is a necessary truth that whatever is water is H₂O, then whenever we have water we have H₂O, even where we strengthen the antecedent. The case passes the antecedent strengthening test: if this is water and in a cup, it is H₂O; if this is water and today is Thursday, then it is H₂O, and so on. But causal claims clearly do not pass the antecedent strengthening test. A may cause B but that does not entail that A would still have caused B had some C also occurred, because C could have been an interferer with respect to A’s causing of B. Thus the striking of a match may have caused it to light but this does not mean that the striking of the match added to a gust of wind would have caused it to light. Because of this failure under antecedent strengthening, we should say that our causal inferences have to use non-monotonic reasoning, whereas for genuine cases of necessity, monotonicity should apply. It is notable that this argument applies no matter how big the cause is: even if it is taken to be the whole world. It remains true that had there been some further factor present, counterfactually, it could have stopped the manifestation.²⁷ Some seek ways of automatically ruling out the addition of a preventer, such as with talk of totality facts or consideration of closest possible worlds only.²⁸ Perhaps that can be done. But what then is the argument that causes necessitate their effects? Isn’t it merely assumed that they do so? We offer a test of necessity. Merely refusing to take the test is not the same as passing it. Propose some other test, if you like, and then we can discuss it; but don’t just take the necessity of causation as the default assumption. This point as articulated thus far can be taken as an external principle of dispositionality: something external to the disposition can prevent it from manifesting, which is sufficient grounds already to conclude that the manifestation was not necessary. A thorough-going dispositionalist, however, can take it also as an internal principle. Even without the presence of an interferer, there could be a failure to ²⁶ Mumford and Anjum (2011a, ch. 3). ²⁸ Mumford and Anjum (2011a, 67–70).

²⁷ Mumford and Anjum (2011a, 64–7).

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manifest. There is, of course, a disposition to manifest. And dispositions can come in degrees of strength from very weak to very strong. But there is no guarantee that any disposition will manifest even if unimpeded. The disposition ‘only’ tends to manifest. Following work with Rani Lill Anjum, this stronger variety of dispositionalism is the one I favour. Dispositionality is thus something less than necessity, but it is also something more than pure contingency. It involves a modality that is intermediate and, we have good reason to believe, irreducible to the two established modal values of possibility—or pure contingency—and necessity.²⁹ The cause disposes towards a particular subset of all the possibilities. There is a tendency towards a certain type of outcome rather than to all the many others that are logically possible. The account is, therefore, antiHumean because it has more than pure contingency; but it nevertheless is a mistake of anti-Humeans to think that a cause must necessitate its effect. One could think of this as a rejection of the principle of sufficient reason. When natural events and processes occur, they don’t do so because they were necessitated. There were dispositions towards them, which became manifest. In addition to this, we can also note that many of the statements that are useful to science are not, in any case, of the universally quantified form. Generic statements, such as ‘metals conduct electricity,’ ‘glaciers form U-shaped valleys,’ and ‘birds fly’ are useful even though they do not entail a corresponding exceptionless regularity.³⁰ Arguably these can take a dispositional reading, too; but they are notable for applying to kinds of thing: metals, glaciers, and birds. As a dispositional law, this can be understood to be about what is disposed to happen for those kinds: what they tend to do. We will see that this provides us with an interesting type of case, effectively ascribing a disposition to a kind rather than to each individual member of the kind.

5. Gravitational Attraction We can illustrate the dispositional account with the example of the law of gravitational attraction, which one would have thought was the paradigm case of a law of nature. The law states the force of attraction between any two bodies is a function of their masses, M₁ and M₂, and their distance apart d. The force F is equal to GM₁M₂/d², which includes a fixed value G, known as the gravitational constant. I will accept this as a truth, despite its use of the notion of force (in contemporary physics forces are no longer invoked, though in their place is something that plays the same role, namely curvature of space-time). Clearly the law is conducive to a dispositional interpretation. It concerns the attraction between two bodies with certain mass properties and a certain distance apart. We can give this an Einsteinian interpretation

²⁹ Mumford and Anjum (2011a, ch. 8).

³⁰ Drewery (2000).

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in terms of space-time if we like. But what, then, of exceptions? I will say that because of the law’s dispositional character, while there is one interpretation under which exceptions can exist, in another sense the law is absolute and exceptionless. The sense in which the law can have exceptions is that as well as the gravitational attraction between the two objects in question, there are also other attractions and repulsions constantly at work. This presents a three-body problem or, more generally, n-body problem. Overall, two bodies may have no inclination towards each other at all. In other words, their attractions to other objects may be stronger than their attractions to each other such that they have no actual movement towards each other. There can be an attraction without an occurrent movement or overall tendency to move because of all the other factors at work. If one could somehow insert a force-measuring device between our two objects, we may well find that there is no overall attraction at all, though this is a thought experiment only. In another sense, however, which I will take to be primary and the intended meaning of the gravitation law, it has no exceptions whatsoever and applies universally to every pair of objects in the universe. The law, I contend, is to be understood dispositionally rather than occurrently. It is about the tendency between the two objects rather than about the manifestation of that tendency. The law tells us nothing about the actual movements of any two objects, except that such movement will be in part determined by this attraction. And even though we do not measure this attraction between two objects in any simple way, we nevertheless accept that it is there, between every two objects, no matter how distant and massive, in the universe. Understood in this dispositional way, there is no reason to think of any exceptions to the law at all. Thus, even though there might be no overall tendency that attracts A and B, this may be thought simply as a resultant power. And this does not entail or require that there is no component power attracting A and B, even if they are overall inclined to move apart. How do we know this component power is there? Consider the counterfactual case in which all other forces upon A and B are removed or, more realistically, cancel each other out. Then the attraction between A and B, being left alone, would be able to manifest itself. One may also consider the other counterfactual: what would happen if the attraction between A and B were removed? Plausibly, this could also make a difference. A and B would move apart even more quickly. Even if the attraction between A and B is interfered with, then, by other bodies, this is no reason to think it in any way falsifies the law of gravitational attraction understood dispositionally. Indeed, this disposition between A and B can be, and is, exactly as the law describes.

6. Mill, Tendencies, and Dispositions A powerful reason why events in our history do not form simple regularities is of course because there are many different laws in operation, as Pietroski and Rey say

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making their own contribution,³¹ sometimes working with each other and sometimes against each other. This is the external principle, described above. Ours is a complex world, as far as we know. But through such complexity and messiness, laws can remain a constant precisely because they are not about what occurrently happens but only about what tends to happen. Mill saw this in his extensive discussion in A System of Logic and understood that laws could counteract each other.³² The best way he saw of understanding this was to accept that laws are about only what tends to happen: All laws of causation are liable to be in this manner counteracted, and seemingly frustrated, by coming into conflict with other laws, the separate result of which is opposite to theirs, or more or less inconsistent with it. And hence, with almost every law, many instances in which it really is entirely fulfilled do not, at first sight, appear to be cases of its operation at all. These facts are correctly indicated by the expression tendency. All laws of causation, in consequence of their liability to be counteracted, require to be stated in words affirmative of tendencies only, not of actual results.³³

This suggests a way of understanding the history of events along the lines of vector composition.³⁴ Individual powers can be understood as component vectors that, through addition, subtraction, or other, non-linear functions, combine to produce a resultant vector. When we measure the force of attraction between any two objects, we can only measure a resultant, which is the force overall produced by all the applicable factors. The individual laws should be understood to concern the component vectors and there are good reasons to be realists about them. Vectors are a way of understanding individual powers or dispositions and these seem really to be there because, for instance, they sustain, or are the truth makers of, counterfactuals. We have good theoretical reasons to believe that if a certain power P were not there, making its contribution to an overall resultant situation, the overall outcome would have been different. If the law of gravitational attraction were removed, for example, we have strong theoretical reasons to think that it would make a noticeable difference to what actually happened. Similarly, in a tug-of-war, two teams pull against each other on a rope. In a sense, there is only a single force being exerted at any one time. But there are good reasons to be realists about the component forces—those exerted by each individual team member—and not just the resultant force. If one team member gave up and left the contest, for instance, it could make a big difference to the outcome. Because laws, as I have argued here, are to be understood as essentially about the component dispositional contributions made to an overall situation, it is very tempting to opt for an ontology of real dispositional properties. Furthermore, if the laws are about those dispositional properties, then it is further tempting to see laws

³¹ See also Cartwright (1999). ³³ Mill (1843/1973, 443–5).

³² Mill (1843/1973, Book III, ch. X, sec. 5). ³⁴ Mumford and Anjum (2011b).

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not as fundamental metaphysical constituents of the world after all. It is the dispositions that do all the work and laws might be reducible phenomena or, one might suggest, eliminable. But I will not pursue this thought any further here.³⁵

7. A Dispositional Reading of ceteris paribus In this section, I want to support a dispositional reading of the ceteris paribus clause, as some others have done before me. But I want to move further than just that idea and consider the question of where this leaves the issue of exceptions. As Lipton argued,³⁶ a dispositional reading is the best way to understand the cp-clause. Others have reasoned along similar lines,³⁷ although doubts have also been raised about the strategy.³⁸ What is useful about the dispositional reading is that we do not have to engage in the task, which might not be completable in any case, of specifying the actual conditions in which the cp-laws hold and the conditions in which they don’t. We do not have to replace the cp-clause with a list of circumstances that it covers. Instead, the cp-clause should be taken as indicating a dispositional reading of the law statement. ‘Fs are Gs, cp’ would mean that the things that are F are disposed—and no more than disposed—to be G. The cp-clause indicates this kind of sui generis, dispositional relationship between being F and being G, which does not entail a strict regularity in actual events. The cp-clause might even be used to indicate something weaker. It is one thing for everything that is F to be disposed towards being G. But there might also be cases, still worthy of consideration as a law, where we do not even have this. Suppose that being F disposes not directly towards G, but only towards having the disposition to be G. This would be a kind of dual disposition: a disposition to acquire a further disposition. Might we not want to allow even this connection a nomological status? It would be a kind of law that could sit with Lowe’s normative account, for which he introduced a new sortal logic.³⁹ Drewery has expressed concerns about the view, and for that reason doubts that dispositions can account for all cp-laws.⁴⁰ I will try to allay her concerns and offer a defence of Lowe’s stance. The basic idea is that it could be a law that ravens are black even if there are some ravens, albinos for instance, that are not even dispositionally black. A raven that is dispositionally black might not be occurrently or actually black, such as one that has bathed in whitewash. This raven has a disposition to be black but does not manifest it. But an albino raven is not even dispositionally black because, presuming that genetic explanation is applicable here, it does not have the genes that dispose towards black plumage.

³⁵ See Mumford (2004a). ³⁶ Lipton (1999). ³⁷ Cartwright (1983, essay 2); Woodward (1992); and Hüttemann (1998). ³⁸ Drewery (2001). ³⁹ Lowe (1982). ⁴⁰ Drewery (2001).

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There are some complex issues here of what it is to be a raven but a suitable account, I claim, could underwrite the idea that being a raven disposes towards blackness even though not every individual raven has the disposition in question. The explanation of this may be partly genetic but could also invoke a notion of ancestry and kind membership. To be a raven is to be a member of some natural kind on the basis of certain characteristics that are historically shared among kind members.⁴¹ Such properties might cluster homeostatically⁴² and include inherited genetics. But again, given that natural causal processes only dispose towards certain outcomes,⁴³ this does not necessitate the possession of the specific blackness gene by any particular descendant of a raven. On a dispositional account of laws, therefore, it could still qualify as a law that ravens are (dispositionally) black even if it is not the case that every kind-member is (dispositionally) black, but only that in virtue of being a raven they are disposed to have the blackness disposition. For this reason, in Lowe’s sortal logic, law statements are not universally quantified at all but are merely disposition ascriptions to kinds.⁴⁴ Drewery questions whether we can make sense of ascribing dispositions directly to kinds, rather than to the individual kind-members.⁴⁵ There is a problem if we cannot. The issue can also be put in Thomistic terms. Thomists draw a distinction between first potentiality and second potentiality.⁴⁶ First potentiality is a power to acquire a power. For example, I cannot speak Norwegian but I do have the power to learn it. If I do learn, then I have the second potentiality, which is a power already acquired and which I can then exercise. I ‘can’ speak Norwegian in the sense that I could learn (first potentiality) even though I ‘cannot’ speak it right now (second potentiality). On this basis, it might be objected to the above theory of cp-laws that because the albino raven is not even dispositionally black (second potentiality), then there is no warrant for saying there is a law that ravens are black, for in some cases, a raven would have first potentiality blackness only. However, this is where the approach of Lowe has its explanatory value.⁴⁷ Law statements should not be interpreted as statements about individual ravens—that they are dispositionally black—over which we universally quantify. Armstrong gives adequate reasons why laws are not directly about that.⁴⁸ Such laws would not sustain counterfactuals, for example, which we think laws should. Lowe’s alternative is to instead conceive of laws as being about what the kind is dispositionally like, rather than members of the kind. Thus, even if it’s not the case that each individual raven is dispositionally black, meaning that not all of them have the second potentiality, the kind as a whole could be disposed towards blackness. What would that mean? Presumably there would be some background theory that can explain it. The theory

⁴¹ ⁴³ ⁴⁴ ⁴⁷

Mumford (2009). ⁴² Boyd (1999). Mumford and Anjum (2011a, ch. 10) and Austin (2015). See also Mumford (2000, 2004b). ⁴⁵ Drewery (2001). Lowe (2006). ⁴⁸ Armstrong (1983).

⁴⁶ Feser (2014, 41).

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might allude to the shared genetic lineage from which a disposition evolved for the passing on of the blackness gene. The empirical details don’t really matter. The point is that there is a lawlike dispositional behaviour found in the raven kind that manifests itself in individual ravens tending to be black, and tending to be dispositionally black. This remains true of the kind even if there are some individual ravens without the disposition. The key commitment was already stated. Laws are about what is disposed to happen: what tends to be. There is a tendency for ravens to be dispositionally black. This is a truth of how the kind is disposed to be. The dispositionalist should side with Lowe and Armstrong on this issue and not succumb to the empiricist temptation to start formulating a law with enumeration of, and quantification over, particulars. Laws can work on a higher level of that, concerning primarily what is true of a universal or a kind. But where does this leave exceptions? As far as I can tell, they vanish altogether and the problem effectively dissolves. Once laws are understood to have this kind of dispositional force only, we can see that, so qualified, they become exceptionless. The ceteris paribus qualification should not be understood as allowing exceptions to the law but as instead indicating its dispositional modality, which does not entail a universal regularity of occurrences. This means that putative exceptions are not in conflict with the law. That Fs dispose towards G is perfectly consistent with there being an instance of F that is not G. In no way does this violate the law. There may be a reason why this individual F is not G. The disposition may have been prevented by the action of other (dispositional) laws, not stimulated in quite the correct way, or just not suitably released. But, according to the internal principle, it might just be that the instance in question of F just happens not to be G, for no particular reason. That is still consistent with there being a tendency of Fs towards G. For something to count as an exception it would have to be a case where F did not dispose towards G, rather than being disposed towards G but just not manifesting it. Given the view of Lowe, we now see that there can be higher-level cases of this concerning how kinds are disposed to be. There might still be some Fs that are not disposed towards G where the law concerns a natural kind tending towards having a certain disposition. In virtue of being an F, it might still be disposed towards being disposed to G. So it would be only where we had an exception to this further dispositional connection that we should start to think in terms of falsification of a law. A putative law would be false if it ascribed a disposition to a kind that the kind did not have. Heated bodies do not give off caloric, for instance, as was one time believed.

8. Conclusion I have suggested that the best way to understand the ceteris paribus clause is to take it as indicative of the dispositional force of the law it qualifies. This means that we do not have to provide specific occurrent content to the clause, but nor does it lapse into

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triviality. A law with dispositional force is only about what will tend to happen in certain circumstances, such as being an F only tending towards being a G. If we understand laws this way, it means that we have to reconsider the place and nature of their putative exceptions. A true law can have no exceptions at all, though it is still, on this account, a cp-law. Indeed, it is because it is a cp-law that it has no exceptions. Certainly, an F that is not a G is not, on its own, enough to count as an exception to a law connecting Fs and Gs that has dispositional force. Some sense can still be made of one notion of exception, but on this account it is a very weak sense. It would be an exception in the sense of something having a disposition but not manifesting it. This hardly counts as a proper exception to a law at all. That something is not occurrently black does not falsify it being dispositionally black and unmanifested dispositions are in any case so very widespread as to be commonplace. The notion of a cp-law can in this way be vindicated. And if all laws have this dispositional character, although I have not here offered a conclusive argument for such a view, it suggests that all laws are cp-qualified. The idea that causal laws in nature have only this dispositional force sits especially well with the neo-Thomist, neo-Millian philosophy of nature that I support.⁴⁹

⁴⁹ An earlier version of this chapter was presented at the University of Rennes in 2004. My thanks to all who gave comments at that event. The final version was written with the financial support of the AHRCfunded Metaphysics of Science project. I am grateful also for the input of the Nottingham dispositions group with whom many of the ideas in this chapter were discussed: Rani Lill Anjum, Charlotte Matheson, Markus Schrenk, and Matthew Tugby. I owe an additional debt to Rani Lill Anjum for always believing in this paper and also for our joint work on the notion of what tends to be, of which this is an application. The chapter is dedicated to the memories of Peter Lipton and Jonathan Lowe.

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12 Are Laws of Nature Consistent with Contingency? Nancy Cartwright and Pedro Merlussi

1. Introduction Are the laws of nature consistent with contingency about what happens in the world? That depends on what the laws of nature actually are, but it also depends on what they are like. The latter is our concern here. Different philosophic views give different accounts of the sort of thing a law of nature is. We shall look at three that are widely endorsed: ‘Humean’ regularity accounts, laws as relations among universals, and disposition/powers accounts. Our question is, given an account of what laws are, what follows about how much contingency, and of what kinds, laws allow? Of the three types we shall look at, powers stand out as especially apt for admitting contingency, or so it would appear from conversations we’ve been engaged in, both with powers advocates and with powers opponents. Our investigation here suggests that this is not so. A powers account of laws may admit contingency but it need not. Conversely, the other accounts may rule out contingency but they need not. In all three cases, we shall argue, the root idea of what laws are does not settle the issue of whether they allow contingency. Advocates of the different accounts may argue for one view or another on the issue, but (at least as we understand the accounts) this will be an add-on rather than a consequence of the basic view about what laws are. Here we explore the possibility of various kinds of contingency in nature, contingency despite the pockets of rough order we observe in our daily lives and of precise order we report in our modern sciences. But, are contingency and order not obviously in opposition? Yes, we think they are . . . if a picture of nature dominant since the Scientific Revolution is correct, that order arises from the rule of universal deterministic laws, laws that hold everywhere and everywhen and that cover all aspects of what happens. But, we shall argue, that picture is not dictated by any of the three kinds of accounts of laws we investigate. Contingency and order are not in conflict on a ‘Humean’ regularity view of laws, as we describe in Section 2. They are also not in conflict if the source of order in nature is relations among universals, as we

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discuss in Section 3, nor if it is powers and dispositions, neither on Alexander Bird’s version nor on that of Nancy Cartwright and John Pemberton. Cartwright and Pemberton argue from how much of modern science works; Bird, by contrast, approaches the issue with the questions and perspectives of metaphysics. We shall briefly review his account in Section 4; in Section 5 Cartwright, following consultation with Pemberton, develops various ways in which contingencies are possible on the view of powers (which they call ‘capacities’) they advance. We shall assume for this discussion that whatever laws of nature are, they are the kinds of things that our current best science might be representing—not that our current science has it right but that we don’t want our philosophic doctrines about what kinds of things laws of nature are to rule out that the world is pretty much as current best science pictures it. In particular, then, we want to admit that candidate accounts of what laws of nature are should allow at least this: whether a radioactive atom decays in some given period of time is contingent, though the probability of this happening is not. Clarifying the question. Whether contingency is possible given the laws of nature depends not only on what kinds of things laws of nature are but also on what contingency consists in. We distinguish several different questions one might ask in asking ‘Do the laws of nature allow for contingencies in nature?’¹ To be as neutral as possible we shall talk of laws covering a happening P if they, along perhaps with what the view of laws on offer counts as the ‘right’ kind of facts (boundary conditions, initial conditions, facts about the past, etc.), say that P will happen (as is typical with common accounts of deterministic laws) or that it is allowed to happen (as with common accounts of probabilistic laws). Using L to label the complete set of correct laws and P₀(L,P) for the additional facts that bring L to bear on P, here are the questions we want to keep sorted from one another. Extent. Is everything that happens covered by L? For instance, there may be happenings, or kinds of happenings, or whole domains about which L is silent. Permissiveness. When L speaks about the outcomes that are to occur, what kind of latitude does it admit? For instance, does it always select a single happening? Does it always lay down at least a probability, or can L admit a set of different outcomes, remaining silent about their probabilities?² Reliability. Does what L (plus some relevant P₀(L,P)) says is to happen always happen? For instance, can there be exceptions to L and yet L still be the correct and complete set of laws? Potency. Do the things that L speaks about happen on account of L? Or, for instance, merely in accord with L?

¹ We should note that we are not concerned with what the actual laws allow but rather with what laws allow by virtue of the kinds of things they are. It may be, for instance, that a particular account allows that laws may be either deterministic or probabilistic but that the actual laws are all deterministic. ² We propose treating laws that say ‘anything goes’ in some circumstance as not covering that circumstance and thus limited in extent.

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Free will. If P, which happens, is an action of a person, is ~P consistent with P₀(L,P) (whatever might be the appropriate P₀(L,P)) obtaining and L being the correct and complete set of laws? We introduce the last question because it hovers in the background. Indeed, it is one of the things that motivated the thinking behind this chapter: Merlussi is otherwise writing on the consequence argument in metaphysics (to be discussed in Section 2), which begins with a version of determinism to argue to the conclusion that nobody ever could have done anything to make P false, for any P that describes human actions; and it is often suggested in conversation with Cartwright and Pemberton that a capacities account of laws like theirs leaves more room for free will than other accounts. Free will is, we all know, a huge question to which over millennia an enormous amount of intense thought has been dedicated, involving of course debate over the very formulation of the problem. Still, we think there are some simple observations we can make about how some accounts of laws of nature bear on the aspect of the question we formulate here. We will not always have much to say about every question with respect to each view of laws we survey but rather focus on what might not be altogether obvious or on where interesting differences lie. We will not address potency seriously at all. It is generally supposed—though not without objections—that universals and powers accounts allow for potency, as well as accounts that involve ‘necessary’ regularities, whereas ‘Humean’ regularity accounts do not. We shall not take up this issue because we have nothing useful to add. We list it for completeness and to make clear that it is a separate issue from the others. There are two guiding ideas we rely on throughout in considering extent and permissiveness. The idea for extent is simple: There may be situations where the laws are silent; they simply do not cover those situations. This is an issue, we claim, that is orthogonal to questions about whether laws are permissive when they do speak. For instance, L may be deterministic in the sense that for each appropriate P₀(L,P), L admits one and only one P to occur, yet limited in extent because some real situations are not P₀(L,P)-type situations for any admissible P₀(L,P), i.e. some situations may not fall into any of the categories for the additional facts that bring L to bear. With respect to extent, a little simple housekeeping is necessary since some of the discussion in both the philosophy of science and the philosophy of religion literature as well as in the related metaphysics literature is confusing (at least to us) because it does not make clear the formulations at stake to begin with, especially with respect to the quantifiers and what they range over. Consider the claim G’: ‘Politeness requires giving an expensive gift to one’s teacher/mentor,’ that we suppose is true in some cultures influenced by Confucianism. Shall we say it is limited in extent, or shall we rather consider G: ‘In cultures A,B,C, politeness requires giving an expensive gift to one’s teacher/mentor,’ which is, we suppose, true everywhere and in that sense not limited in extent? Similarly, Cartwright has suggested that what we think of as the

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usual laws of physics L’ may well be limited in extent in a very specific way: They may be unable to represent all the possible causes of the effects they represent; their truth may then be restricted to just those cases where only causes they can represent are at work.³ Thus, it should more perspicuously be formulated something like this. L: ‘So long as all of the causes of the consequences represented in L’ are features represented in the antecedents of L’, then L’.’ One could think of formulating the issue in terms of domain restrictions: Are these restrictions included in the laws themselves or not? The problem is that it can be difficult to formulate criteria for what counts as a restriction on the domain of a law versus what counts as a feature that it genuinely covers. This is why we formulate the issue as we do: Are there things that happen that the complete and correct set of laws does not cover? As to permissiveness, although we turn to capacities last, it is useful to foreshadow one of the topics discussed there because it will help with understanding our remarks about permissiveness throughout. Cartwright has long urged that some events, even ones in the purview of laws, may just happen—by hap—without even any probabilities assigned by nature. An earring back is stuck in some debris in the crack between the floorboards. You try to lift it with a magnet. The magnet pulls upward on the metal object with a fixed strength and gravity pulls it down with a fixed strength. These activities are both properly treated as sources of forces, where by ‘properly’ she means that there is a general way to ascribe forces for both. There is a magnet and there is a rule in physics for what forces magnets exert; and there is a large mass—the earth—and there is a rule for what force a mass exerts. There is also debris that inhibits the motion of the earring back. Maybe there is another description of this particular debris for which there is a proper rule in physics that assigns a force. But certainly not under the description ‘debris.’ And maybe there is no other such description. We may grant that some causes of motion are forces in the proper sense of that concept but that does not imply that all are. To assume there must be because the debris can affect the motion of the earring back is to make a massive metaphysical assumption beyond the empirical evidence, Cartwright argues. If we leave the issue open, then a new possibility for contingency arises. There is a rule for what force is exerted when the magnet and the earth act together in this arrangement, and on this rule only one resultant force is allowed. But what about the motion of the earring back? Is there a rule that says what one motion will happen in this arrangement when the resultant force of the earth and the magnet acts on the ear-ring back simultaneously with the inhibiting power of the debris, or if not a rule dictating one single outcome, is there a rule that dictates a set of outcomes with a probability measure over them? We have insufficient reason to assume there is, Cartwright has argued, so that assumption should not be forced by our account of what laws are; the account should

³ Cartwright (1989, 2009, 2010).

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leave the question open. Yet surely there is some kind of rule since we have what do seem well-warranted beliefs that the earring back will not fly away at near the speed of light, and also, as Keith Ward⁴ has pressed, that it will not turn into a pumpkin. This is very underexplored territory. But it seems that here may lie yet another source of contingency; we have labeled this permissiveness: When L applies, given a relevant input P₀(L,P), L might admit only one outcome, in which case L is not permissive. On the other hand, L may be permissive in that L admits a set that includes more than one outcome, and in the latter case, L may or may not provide a probability over that set.⁵

2. The ‘Humean’ Regularity Account The central motivating idea behind what we shall call the ‘Humean’ regularity account of laws is not about laws but about the make-up of the world. The facts that constitute the world involve only qualities, quantities, and relations that are occurent, where ‘occurent’ means different things to different philosophers who call themselves ‘Humeans.’ What they all have in common is that they want to exclude any kind of ‘modal’ features. There are no causings, no necessitatings, no doings, no making-things-happen-ings. In answer to the question ‘What is it to be a law of nature?’ the naïve ‘Humean’ account states that laws are regular associations among occurent features. But this is thought to be problematic. There are true accidental regularities that are not laws, it is supposed. To use Hans Reichenbach’s memorable example, ‘All gold spheres are less than a mile in diameter’ is a genuine regular association, but this does not seem to be a law.⁶ So, it is commonly assumed, a satisfactory ‘Humean’ view of laws should distinguish laws from accidental regularities. This is what David Lewis’s best system account (BSA) sets out to do.⁷ Since BSA is very well developed and widely adopted, we shall focus on this version of the ‘Humean’ regularity account. However, the main arguments we put forward should go through for any acceptable ‘Humean’ account of lawhood, including Craig Callender and Jonathan Cohen’s ‘Better Best Systems Account.’⁸ In Counterfactuals and ‘Humean Supervenience Debugged,’ Lewis takes as a starting point a short note written by Frank Ramsey in 1928.⁹ Lewis’s restatement of Ramsey’s passage asserts that ‘a contingent generalization is a law of nature if and only if it appears as a theorem (or axiom) in each of the true deductive systems that achieves a best combination of simplicity and strength.’¹⁰, ¹¹ ⁴ Personal conversation with respect to Cartwright and Ward (forthcoming). ⁵ Clearly this supposes some already given way to individuate outcomes. ⁶ Reichenbach (1947, 368). ⁷ Lewis (1973). ⁸ Cohen and Callender (2009). ⁹ Lewis (1973, 73) and Lewis (1994, 478). ¹⁰ Lewis (1973, 73). This looks like a use/mention confusion but it is almost certainly harmless. We shall try to avoid confusing the two but occasionally for ease of expression we will follow Lewis in talking in the formal mode when the claim is really one in the material mode. ¹¹ Here’s what this means: Consider a true deductive system in which the general claims that represent laws of nature appear as a set of true sentences T that is deductively closed and whose non-logical

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Notice two important features of this view. First, laws supervene on the particular matters of fact. This is so because laws merely summarize facts. So, as to potency, laws do not ‘govern’ the world, they are just special regularities that encompass a good many other regularities. The particular matters of fact determine the laws of nature in the sense that if the laws of nature are different, that’s because the facts are different. Because of this, the BSA preserves the alleged intuition that the laws of nature are metaphysically contingent, at least so long as it is metaphysically contingent what the facts are. Given this brief description we can look at how the BSA deals with questions of whether L is compatible with contingencies. Extent. Does L cover everything that happens? Following John Earman, one might formulate the question as follows: Do laws have an unrestricted range in space and time?¹² As Earman points out, to deny that laws have an unrestricted range in space and time boils down to saying that there is ‘a region of space-time Ro such that, as far as L is concerned, “anything goes” in Ro.’¹³ More precisely, where M denotes the set of all models of the putative law sentences, this may be formulated as the question of whether claims representing the complete and correct set of laws L satisfies the following condition: (U) There is no non-empty, proper subregion R₀ of space-time such that for any M 2 M, there is an M’ 2 M where M’ ⊨ L and M’|Ro ≈ M|Ro. This condition states that L is valid on a model that is not restricted to some spatiotemporal region, that is, L is ‘universal.’ Given the BSA there is motivation for thinking that the laws of nature should be ‘universal.’ If the range of the axioms (or theorems) of the best deductive systems were limited to some spatio-temporal region, then one would expect more axioms to summarize the whole history of the world. That is, one would need more axioms to cover all spatio-temporal regions. But if the range of the axioms is not limited, then one can naturally expect fewer axioms to summarize all the particular matters of fact. Furthermore, this will not reduce the system’s informativeness, since the axioms now are not restricted to some spatiotemporal region. On the other hand, if nature is fairly unruly outside a given range, adding piecemeal information about what happens there to any set of axioms may vocabulary contains only predicates that express occurent properties. There are many ways systems can be axiomatized. If the axioms of T preclude more possibilities than T, ‘then T is stronger than T.’ Likewise, some true deductive systems can be axiomatized more simply than others, in the sense that they have fewer axioms. The general claims representing the laws of nature will belong to all the axiom systems with a best combination of these two virtues, simplicity and strength. ¹² Earman (1978, 174). ¹³ Earman (1978, 174). We use this formulation because readers may be familiar with it. But there is no reason to assume that nature thinks in terms of space-time regions rather than, as in Cartwright’s view, in terms of what features obtain. For instance, as we noted, her rendering of boundaries on the range of a theory T is roughly this: Those instances of effect E that T covers are the instances for which some or all of the causes of E fall under concepts available in T.

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increase informativeness at too great a cost to simplicity. So the laws may be limited in extent. Despite the fact that the BSA explains why we might expect the laws of nature to be universal, it seems that Earman is right in saying that there is no a priori guarantee that the laws of nature according to the BSA will satisfy (U).¹⁴ Permissiveness. Within its domain, under the BSA, does the correct L (plus relevant initial or boundary conditions) always single out a unique outcome? In order to answer this question one needs to bear in mind the main motivation behind Humeanism about laws. The world is void of modalities—no causings, no necessitatings, no probabilifyings; the world is nothing but a mosaic of occurent events. Laws summarize what happens in this mosaic, rather than ‘governing’ what the particular matters of fact are. If L is deterministic, given an appropriate P₀(L,P), L admits only one outcome. But there is nothing in the Humean motivation that makes determinism natural. The best summary may be provided by purely probabilistic laws or by laws that constrain outcomes to a given set but do not choose among them nor lay a probability over them. As Helen Beebee points out, whether the world is best axiomatized under deterministic laws depends on how regular the world is.¹⁵ The world can be modality free and still irregular enough to be summarized best by non-deterministic laws. Reliability. So as not to muddle together issues of extent, permissiveness, and reliability, let’s consider the most difficult case for contingency in the reliability sense: where the laws have universal extent and are deterministic, allowing only one output for any relevant input. It looks at first sight as if in this case on the BSA, they must be reliable. There can be no exceptions to the correct laws. We think, however, that there is still some wiggle room and will offer two ways that might be thought sympathetic to the ‘Humean’ viewpoint that might allow for exceptions, one of which is due to Lewis himself. For the sake of this discussion we propose to adapt Earman’s definition of determinism in terms of possible worlds to define deterministic laws because it makes for a ready connection to the Lewis wiggle. Let L stand for ‘L is the correct set of laws,’ then define ‘deterministic’ thus: Laws L are deterministic iff for any P that L covers and any P₀(L,P) that is ‘appropriate’ input to L for P and any logically possible worlds w, w’ in which L, if w and w’ agree on P₀(L,P), they agree on whether P obtains. An interesting way to address the question of the reliability of deterministic laws under the BSA is by considering Scott Sehon’s objection to the standard definition of determinism.¹⁶ First, we start by pointing out that D: If L and L is deterministic then for any P that occurs and that L covers and any P₀(L,P) that occurs that is an appropriate boundary/initial condition for P with respect to L, ▫((P₀(L,P) & L) ⊃ P). ¹⁴ Earman (1978, 180).

¹⁵ Beebee (2000, 575).

¹⁶ Sehon (2011).

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To see why, suppose that L is deterministic and P₀(L,P) is an appropriate boundary/ initial condition for P with respect to L and P. Let W stand for the collection of all possible worlds. Consider an arbitrary world w in W where P₀(L,P) and L. Because L is deterministic, if P obtains in any world where L and P₀(L,P) obtain, it holds in all worlds where P₀(L,P) and L obtain, including w. P obtains in our (the actual) world where L and P₀(L,P) obtain. So P obtains in w and thus (P₀(L,P) & L) ⊃ P in w. Since w is any arbitrary possible world, ▫((P₀(L,P) & L) ⊃ P) follows. Sehon, however, thinks that this is problematic. He argues that L and P₀(L,P) should allow exceptions. Even if the correct laws are deterministic, Sehon claims, it should be logically possible that there is, for example, an interventionist God (IG) that could miraculously change water into wine.¹⁷ As Sehon says, ‘necessarily, if an IG exists, then it is possible that the same initial state of affairs obtains, along with the same laws of nature, and yet P is false’—i.e. it is possible that P₀(L,P)&L&~P.¹⁸ His reasoning can be spelled out as follows (using IG to stand for ‘There is an interventionist God’): 1. 2. 3. 4.

▫(IG⊃◊(P₀(L,P)&L&~P)) Premise. ◊IG Premise. ◊(P₀(L,P)&L&~P) From 1&2, assuming S4. ~▫((P₀(L,P)&L)⊃P) From 3.

And (4), clearly, is the contradictory of ▫((P₀(L,P) & L) ⊃ P), which follows from the assumption that L is deterministic. Note that Sehon’s main point does not depend on the premise that an IG is logically possible. One might try to cast Sehon’s objection as a call for a domain restriction: L holds everywhere that there is no interventionist God (L holds if ~IG). ‘Humeans’ might not like this because there is no way that the domain restriction could be brought into the antecedents in the laws of nature since laws are supposed to involve only occurent features, and God’s intervening does not seem a good candidate for an occurent feature on any ‘Humean’ account of ‘occurent’ we know. That aside, the problem is that determinism would be incompatible, say, with the logical possibility of an interventionist demon, in the sense that, necessarily, if an interventionist demon exists, then it is possible that P₀(L,P)&L&~IG&~P. So, Sehon’s main worry is not about the logical possibility of an IG, nor about the logical possibility of a demon in particular. It is about the logical possibility of the laws of nature being violated.¹⁹ Thus, Sehon urges, exceptions to what L (and P₀(L,P)) say should happen should be possible even if determinism is true, precisely because it must be logically possible to violate the laws. And if the BSA does not accommodate that, there must be something wrong with the BSA as an account of laws. ¹⁷ Sehon (2011, 31). ¹⁸ Sehon (2011, 31). ¹⁹ A domain restriction in this case seems to make law claims tautological, which they should not be for the Humean: ‘As are regularly associated with Bs except when they aren’t.’

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In what follows, we will show how a Lewisian might reply to this argument, showing that the BSA may be consistent with assuming that the correct laws are deterministic and yet can be violated, at least in a sense. The task then is to show that these two propositions are consistent: p: The correct laws L are deterministic. q: It is possible to violate (the correct laws) L. The first strategy is to hedge on p, using Lewis’s own notion of soft determinism, which is supposed to allow a sense in which agents are able to do things such that, if they were to do them, what L says happens does not happen.²⁰ Let us assume the truth of p and thus of D, so that some statement about the distant past, P₀(L,P), and L logically imply, for instance, P: ‘Agent A did not raise her hand.’ What if A had raised her hand? There are three options: 1. If A had raised her hand, contradictions would have been true. 2. If A had raised her hand, P₀(L,P) would be false. 3. If A had raised her hand, L would be false. Someone like Lewis will naturally reject option 1. Even if the agent had raised her hand, contradictions would not have been true. Lewis also denies 2. Even if the agent had raised her hand, the past would still be the same, so P₀(L,P) would still be true.²¹ Thus, if we want to say that the correct set of laws L is deterministic and sometimes we are able to act otherwise, the only option remaining consistent with Lewis’s viewpoint is 3. Thus, given P₀(L,P) and D, ~P implies L is false. Yet, we are supposed to be arguing that L are the correct laws. How is that possible? Following Lewis the clue is: correct in what worlds? To see how this works we need to draw a distinction between two senses in which one can violate a law: Weak sense: An agent is able to do something such that, if she were to do it, a law would be violated, either a law of the actual world or a law of nearest possible worlds. Strong sense: An agent is able to do something such that, if she were to do it, a law would be violated and this law would be of the actual world. For example, in the weak sense, if the agent were to have raised her hand (i.e. we assume she did indeed raise her hand in the actual world), contrary to what L says, then L would have been violated before the hand raising. To use Lewis’s phrase, a ‘divergence miracle’ would have happened before that, that is, there would be a violation of the laws of nature that hold at our actual world, and this violation would not be caused by A’s action. Note that to say that there is a violation of the laws of nature in the weak sense is not to say that the violated laws are the laws of the same

²⁰ Lewis (1981, 114).

²¹ Lewis (1979a).

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world where they are violated. The term ‘miracle’ is used to express a relation between different possible worlds. As Lewis says, ‘a miracle at w₁, relative to w₀, is a violation at w₁ of the laws of w₀, which are at best the almost-laws of w₁.’²² So with a divergent miracle in our actual world, whose laws are the ‘almost’ laws of a nearest world where L is not violated, we can violate the correct laws of that nearby world. Or vice versa. Now, if by ‘violating a law’ we mean the weak sense where what we violate is an ‘almost law,’ not a real law, of our world, then it seems agents may be able to violate laws that are deterministic. But what if by ‘violating the laws of nature’ Sehon means the strong sense? The strong sense is the one in which the laws are violated in the actual world that are the laws of the actual world. This seems what Sehon has in mind when he says that, if IG, then it is possible that we have the same laws, the same past, and yet P is false. However, if by ‘violating a law’ Sehon means the strong sense, then someone like Lewis will deny that it is logically possible to violate a law in the strong sense. This is so because, as Lewis says, ‘any genuine law is at least an absolutely unbroken regularity.’²³ Given the BSA, it is clear why we cannot violate laws in the strong sense. Suppose it is a law that no object moves faster than light. If someone were to throw an object that moves faster than light, then that law would not be true. Since Lewis’s ‘Humean’ laws are true regularities, if it is a fact that a certain stone moves faster than light, then it cannot be a true regularity that no objects travel faster than light. ‘Humeans’ might, however, consistent with the commitment that there are only the occurent facts of which laws are summaries, take a more instrumentalist line. The best summaries may not be required to be true, especially if this brings about a big gain in simplicity. They could admit of exceptions but be right most of the time. Or they could be wrong most all the time yet still very nearly right most, even all, of the time. This is like William Wimsatt’s view that laws could be templates that fit widely but in many cases not exactly.²⁴ Whether admitting false claims as the correct laws is a good idea on the ‘Humean’ view depends on what the world is really like. Cartwright has argued that high-level laws in physics often get fitted to the real details of real situations only by adding ad hoc corrections.²⁵ That could be because we have just missed out on the factors that support those corrections and that bring the situation genuinely under the laws. But it could be that that is just what the world is like. There is no single uniform pattern but only a template which fits widely but not very exactly. If the latter is the case, the BSA can be maintained while allowing contingency in the reliability sense, so long as the demand is given up that the best summary of the facts be true.²⁶

²² Lewis (1979a, 469). ²³ Lewis (1981, 114). ²⁴ Wimsatt (1992). ²⁵ Cartwright (1983). ²⁶ Here it is easy to make things look simpler than they are by blurring use/mention distinctions. If laws are ‘false’ but ‘nearly true’ then the laws will not be facts as we first claimed for BSA but rather only very similar to facts.

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Potency. Do the things that L speaks about happen on account of L? Or, for instance, merely in accord with L? Perhaps this is the least problematic question to answer according to the BSA. Clearly, the things that L speaks about happen merely in accord with L. Free will. The question whether the ‘Humean’ account of laws plus assumption D, which follows from the hypothesis that the correct laws are deterministic, is compatible with the possibility of agents doing other than what they do, can be introduced in the context of the currently central argument for incompatibilism,²⁷ namely the consequence argument.²⁸ One of the crucial premises of the consequence argument is that the laws of nature are not up to anyone. The modal formulation makes use of a modal sentential operator ‘■’ in which ‘■P’ abbreviates ‘P and no one has or ever had any choice about whether P.’ ‘■’ is supposed to satisfy these two inference rules: (α) ▫P ‘ ■P (β) ■(P ⊃ Q), ■P ‘ ■Q. Here is the consequence argument, supposing that L are the correct laws, the correct laws are true, P is something that happens, and P₀(L,P) is the relevant feature to fix P given L: 1. ▫ ((P₀(L,P) & L) ⊃ P) from determinism. 2. ▫ (P₀(L,P) ⊃ (L ⊃ P)) from 1. 3. ■ (P₀(L,P) ⊃ (L ⊃P)) from 2 and rule (α). 4. ■ P₀(L,P) premise, fixity of past. 5. ■ (L ⊃ P) from 3, 4, and rule (β). 6. ■ L premise, fixity of laws of nature. 7. ■ P from 5, 6, and rule (β). Is premise (6) true? One might interpret the ‘■’ operator in a more precise way as follows:²⁹ (■-def.):

■P if and only if P & ~9x 9α [Can(x, α) & (Does (x, α) ▫! ~P)]

where ‘▫!’ stands for the counterfactual conditional, x ranges over agents, and α ranges over all past, present, and future action types. The idea is that there is nothing that anyone can do such that if they were to do it P would be false. Now, is ‘■L’ true according to this interpretation? As the interpretation above makes explicit use of a counterfactual conditional, and since we are interested in seeing how the ‘Humean’ might answer this question, the natural way to

²⁷ Incompatibilism here understood is the view that if determinism is true, there’s no free will. ²⁸ Cf. Ginet (1983) and van Inwagen (1983). The Consequence Argument is so called because it relies on the consequences of the laws of nature and the past in order to establish incompatibilism. ²⁹ Pruss (2013).

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proceed is to use an account of counterfactuals that is in line with the BSA. So we will presuppose Lewis’s own semantics. At a first approximation, let us say that (C-L): A ▫! B is (non-vacuously) true in a world w iff B is true in all the worlds in which A is true that are closest to w. Given (■-def.), if ‘■L’ is false, then some agent s is able to perform an action a such that, if s were to perform a, then L would be false. To put it in a different way, is L true in all the closest worlds in which P: Agent s does action a? Suppose ~P, that s does not perform a in the actual world w₀. Now, suppose worlds in which s performs a have the same laws as the actual w₀ and these laws are deterministic. Can we consider these worlds to be the closest relative to w₀ among the worlds where s performs a? Since worlds in which s performs a do not agree on P they cannot agree on any P₀(L,P) that with L determines P nor on any R(L,P₀(L,P)) that with L determines P₀(L,P), and so forth. Now L—the set of complete and correct laws of our world—may be very limited in extent. Perhaps they only cover P, in which case the only fact besides P on which these worlds disagree with the actual is P₀(L,P). But this won’t work if they are to account in the way we usually expect for the amount of order we see in the world. For instance, what about all the knock-on effects from all the initial or boundary conditions that are related under L to P? And the knock-on effects of the Rs that need to be different when all the laws in L are deterministic to ensure the P₀s are? When so much divergence from the actual world, w₀, occurs in these worlds, can these worlds be the closest worlds relative to w₀ in which s does a? Following Lewis’s own maneuvers in cases like this, the better option, it seems, is to regard as ‘closest’ those worlds that are just like w₀ up to about the time that s performs a, and then diverge by a divergence miracle. Therefore, the closest worlds in which s performs a are not worlds in which the same laws L obtain. Therefore, the ‘Humean’ who follows Lewis has motivation for considering ‘■L’ false and, consequently, for rejecting premise 6 of the consequence argument.

3. Laws as Relations among Universals Fred Dretske, Michael Tooley, and David Armstrong developed a rival approach to the BSA.³⁰ In what follows our presentation will focus on Armstrong’s view. Laws of nature, according to Armstrong, are necessary relations among first-order universals. The ontological component of a law according to the BSA is a regularity; on Armstrong’s view, it is a second-order relation between first-order universals. Suppose that all Fs are Gs and that the laws of nature ensure this. F-ness and G-ness are taken to be first-order universals. Armstrong states that a second-order contingent relation holds between these two universals. He labels this relation as ‘nomic ³⁰ Dretske (1977), Tooley (1987), and Armstrong (1983).

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necessitation’ and he uses ‘N’ to refer to it. Armstrong symbolizes the relation of necessitation between F and G as ‘N(F,G).’ He also claims that the holding of N entails the corresponding generalization. If the second-order relation N holds between the first-order universals F and G, then ‘N(F,G)’ entails ‘All Fs are Gs.’ On the traditional Armstrong/Tooley/Dretske view it seems that laws are reliable—what they say goes, goes. At least this is the case under the assumption at the core of the view³¹ that the relations that obtain between universals make true the corresponding relations between instantiations of those universals in the real world; what happens in the empirical world depends on and must be in accord with what relations hold among universals. This also ensures that laws are powerful—things happen because they say so. So potency is assured as well. On extent, perhaps the issue is more open. Individual advocates may argue that laws govern all that happens. But that seems to be an add-on to the two assumptions that seem central to the account that, first, laws are relations between universals, and second, any instances of universals that figure in the laws must reflect in the appropriate way the relations among those universals. These do not by themselves imply that every feature that occurs in the world instances a universal that has such relations to others and hence the two do not seem to imply that everything that happens is in the purview of laws of nature. Even if one supposes that it makes no sense to think of features that do not fall under universals, there is still the issue of whether the associated universals all participate in the kinds of relations to one another that make for laws of nature. Permissiveness may also be more open on the laws-as-relations-among-universals view than it seems at first sight. For there may be more relations among universals than just the one—labeled ‘N’—that is the truth maker for the necessitation aspect of law claims. Some universals may be taller or more beautiful than others, which may be irrelevant to what happens in the world when these universals are instantiated. Even among world-guiding relations, necessitation may not be all there is. After all, the view presumably does not want to rule out that a probabilistic theory like quantum mechanics can be correct. One way to allow for this is to keep only N and then suppose that the universal represented by the quantum state is N-related to a universal that we represent by a probability measure. Instantiation of this last seems troublesome though; moreover probability itself, as van Fraassen argues, may best be seen as a modal notion.³² So, in keeping with the view that modalities reflect facts about universals and their relations, another idea for how to handle probabilistic laws is to assume there is another kind of modality beyond that responsible for necessity: ‘probabilifies,’ with various ways to develop this idea further. Key though is that if the universal corresponding to ³¹ Though it has frequently been objected that it is hard to see how this assumption could be true (van Fraassen 1989). ³² Van Fraassen (1980).

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A probabilifies the universal corresponding to quantity Q in accord with Prob (Q ¼ q), then instances of A will be associated with instances of values of Q in a pattern reflecting Prob (Q ¼ q). This leads readily to admitting permissiveness of the kind we see in the capacities account of laws. Once more world-guiding relations are admitted than N, there seems no good reason to suppose that an even weaker modal notion than ‘probabilifies’ may obtain, one that constrains the values Q may take when A is instantiated to a given set but which dictates no particular pattern to them. One or another in the set must be instantiated but which on any occasion is mere hap, with not even a nice probabilitylooking pattern to emerge in the long run. This may, at first sight, seem counter to the universals account of laws. After all, wasn’t the point to find some location for necessity? We think not. The point is to find a location for modality. Universals are introduced in order to enable laws to do a number of jobs. They are supposed to support counterfactuals, to explain why things happen in the orderly way they do, to justify our inductive practices. All this may require modality but other modalities than necessity can do the jobs required. How is it on this view that the laws of nature explain that all Fs are Gs and justify our inductive practice of predicting that the next F we encounter will be G on the basis of past observations that Fs are Gs? It is because the universal associated with F is N-related to that associated with G. But it is not the N-ness of the relation that matters; it is rather the two-fold fact that this relation holds between the universals in Platonic heaven, and whatever world-guiding relations occur in Platonic heaven must be reflected in the behavior of their instances in the empirical world. Other kinds of patterns in the world could then be equally explained and supported by other relations between universals, for instance ‘F probabilifies Q ¼ q to degree p,’ where the p values for Qs satisfy the probability calculus; or F φ-necessitates Q, which is reflected in the fact that Fs are always followed by some value or other of Q in φ.

4. Dispositions, à la Alexander Bird So far we have mainly focused on Lewis’s and Armstrong’s accounts. Although they can both be seen as figureheads for rival camps concerning the laws of nature, Alexander Bird interestingly notes that the accounts have two theses in common.³³ They both take (i) laws of nature to be metaphysically contingent, and they both take (ii) properties to be categorical. Dispositional essentialism (DE) has emerged as an account of laws that explicitly rejects these two assumptions. First, according to DE, the laws of nature are metaphysically necessary, for reasons that we will see soon (though we shall have very little to say about what is supposed to be meant by ‘metaphysically necessary’). Second, DE takes at least some—maybe all—natural ³³ Bird (2005, 2007).

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properties to be essentially dispositional. We will briefly discuss in this section Alexander Bird’s version of DE for a concrete illustration. Similar results with respect to contingency hold for many other versions, making appropriate adjustments. First, Bird adopts the conditional analysis of dispositions (CA). Where D is a dispositional property, S(D) is a stimulus property appropriate to it and M(D) is its manifestation property, (CA) may be symbolized as follows: (CA) Dx $(S(D)x ▫! M(D)x). As Bird points out, (CA) does not merely provide an analysis of the concept D; instead, it characterizes the nature of the property D. Thus, as Bird says, (CA) is metaphysically necessary: (CA▫) ▫M(Dx $ (S(D)x ▫! M(D)x)). Second, DE endorses the view that at least some fundamental properties are essentially dispositional. To say that a property P is essentially dispositional is to say that, necessarily—in the metaphysical sense—to instantiate P is to possess a disposition D(P) to yield the appropriate manifestation in response to an appropriate stimulus: (DEP) ▫M(Px ! D(P)x) Here is how to explain ‘the truth of a generalisation on the basis of the dispositional essence of a property’ (Bird 2007: 46): 1. 2. 3. 4.

▫M(P(D)x ! (S(D)x ▫! M(D)x)) from CA▫ and DEP. P(D)x & S(D)x assumption. M(D)x from 1 and 2. (P(D)x & S(D)x) ! Mx from 2–3.

Since one can generalize over the unbound variable x, we get from 4 5.

8x ((P(D)x & S(D)x) ! M(D)x.

Hence, a universal generalization follows from (CA▫) and (DEP). Furthermore, since both (CA▫) and (DEP) are metaphysically necessary, this generalization is metaphysically necessary as well. It looks then as if any laws underwritten by dispositional properties will be totally reliable, and on Bird’s view it seems that these are all the laws there. The problem with this, though, is that (CA) is often false, Bird notes, because of the existence of finkish dispositions and antidotes.³⁴ However, he argues, rather than ³⁴ ‘An object’s disposition is finkish when the object loses the disposition after the occurrence of the stimulus but before the manifestation can occur and in such a way that consequently that manifestation does not occur’ (Bird 2007: 25). See also Martin (1994) and Lewis (1997). Bird also points out that one cannot eliminate all counterexamples to (CA!) by excluding finks (Bird 2007: 27). ‘Let object x possess disposition D(S,M). At a time t it receives stimulus S and so in the normal course of things, at some later time t’, x manifests M’ (Bird 2007: 27). An antidote or mask to D(S,M) is something that ‘has the effect of

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being a disadvantage for dispositionalism, this is one of its virtues, since the falsity of (CA) allows the dispositionalist to account for ceteris paribus laws. We can just replace the left-to-right implication of (CA) by (CA! *) Dx ! (S(D)x & finks and antidotes to D are absent ▫! M(D)x). Now we deduce the following regularity: 8x (finks and antidotes to D are absent ! ((Dx & S(D)x) ! M(D)x)). This is how the dispositionalist can account for ceteris paribus laws—supposing that in all correct ceteris paribus laws, the conditions that are referred to in the ceteris paribus clause genuinely are either finks or antidotes to the disposition referred to. Conditioning on the absence of finks and antidotes gets built right into the laws themselves. Reliability, it seems, is thus restored, at least for ceteris paribus laws where all that is missing to render the ceteris paribus clause explicit is reference to finks or antidotes. Moreover, Bird also argues that there is a fundamental level of laws where no finks occur and where antidotes are very unlikely.³⁵ In that case, as above, reliability is assured by (CA), as already noted. What then about permissiveness? It seems that where they speak—which seems to be whenever a dispositional property obtains and there are no finks or antidotes to it—DE laws allow only one outcome, the manifestation associated with that disposition. So DE laws seem impermissive. On the other hand, there seems to be nothing in the basic motivations for this account that implies that the manifestation must be limited to a single choice rather than a set of choices, with or without a probability over them. So impermissiveness seems an add-on for DE laws, just as it is for laws when taken as relations among universals or on the BSA. Extent too seems to fare just the same as in the other two accounts so far surveyed, except perhaps limitations on extent are to be expected here, at least so far as the basics we have presented go. The issue is whether everything that happens is a manifestation of (some combination of) essentialist dispositions. Two ways they may not be are immediately evident. First, if not all properties are DE properties then DE laws that supervene on DE properties and their associated dispositions will not cover them.³⁶ Second, DE laws derived above are, as remarked, ceteris paribus laws, which cover only situations where no finks and antidotes obtain. What happens when these do? Or—more to the point—will finks and antidotes always be constituted by essentialist dispositional properties so that what happens when they obtain breaking the causal chain leading to M, so that M does not in fact occur’ when applied before t’ (Bird 2007: 27) See Bird (1998). ³⁵ Bird (2007, 63). ³⁶ Some proponents of DE might, however, hold a mixed view according to which some fundamental properties are essentially dispositional and others are categorical, and so a DE law could connect a disposition with a categorical property. As a result, extent may be retained since laws won’t supervene only on DE properties. Thanks to the editors for pointing this out.

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is then covered by the universal generalization that supervenes on the dispositions associated to those? If not, then DE laws won’t cover everything that happens. So DE laws may well be limited in extent. Recall though that Bird maintains that there is a level of fundamental dispositional properties that are not subject to finks and are seldom subject to antidotes. Does this imply that the correct set of laws covers all that happens? Supposing we substitute ‘never’ for ‘seldom,’ the answer is ‘yes,’ if a kind of total reductionism holds in which everything ultimately is covered by laws deriving from fundamental dispositional properties. But this kind of reductionism does not seem to follow from the basic motivating ideas of a DE account of laws. As with many of the other assumptions we have discussed, it is just an add-on. The real issue for extent then depends on two things. First, are all properties, including those that feature in finks and antidotes, essentially dispositional? And second, are all complexes of properties—like: ‘P(D) and the properties that characterize antidote A to P(D) and fink F to P(D)’—themselves essentially dispositional properties and hence properties that give rise to laws that can cover every case? Suppose the answer to both is ‘yes.’ Is that an add-on or rather a central part of the DE view? The answer here seems less clear than in many of the other cases we have considered and we won’t take a view. But if the answer is yes and this is not deemed an add-on, then DE laws will be, by their nature, universal in extent. What about reliability? Again let’s look at what seems to be the hardest case— where the laws are deterministic, which is where much of the current philosophy of religion and metaphysics literature focuses. As we saw before, if the correct laws L are deterministic, then ▫((P₀(L,P) & L) ! P). This is true also for Bird’s account. But the main difference between Bird’s view and the BSA is how they reply to Sehon’s objection. If Sehon is right, then determinism should be compatible with ‘IG’ being logically possible. However, it should be noted that, in Sehon’s argument, he reads boxes and diamonds as logical necessity and possibility. Thus, his reasoning is only relevant if the box of ▫((P₀(L,P) & L) ! P) is read as logical necessity. It will be clearer if we present his reasoning again. Let ‘▫L’ and ‘◊L’ respectively stand for logical necessity and possibility. 1. 2. 3. 4.

▫L(IG⊃◊(P₀(L,P)&L&~P)) Premise.

◊LIG Premise. ◊L(P₀(L,P)&L&~P) From 1&2, assuming S4. ~▫L((P₀(L,P)&L)⊃P) From 3.

As we can see, 4 implies the contradictory of ▫((P₀(L,P) & L) ! P) if the box is read as logical necessity. Now, if we take the initial or boundary conditions that feed into laws to be facts about the past, which is one typical choice for them, then Helen Beebee can help us think about the issue of logical necessity for ‘Humean’ views: ‘For the Humean, the laws and the current facts determine the future facts in a purely logical way

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[our emphasis]: you can deduce future facts from current facts plus the laws. And this is just because laws are, in part, facts about the future.’³⁷ So, if the BSA is correct, then it should follow from determinism that ▫L((P₀(L,P) & L) ! P), as indeed it does under the definition we adopted in Section 2. That is, according to the BSA, determinism is incompatible with ‘IG’ being logically possible as possibility is characterized by Sehon. On the other hand, if DE is correct, then it seems that determinism could be compatible with ‘IG’ being logically possible even as characterized by Sehon. This is so because the dispositionalist needs only one genuine notion of necessity that applies to issues about what happens in the world, which is metaphysical necessity (Bird 2007: 48). And metaphysical necessity is distinct from logical necessity. As a result, the box of ▫((P₀(L,P) & L) ! P) should be read as metaphysical necessity. Let ‘▫M’ stand for metaphysical necessity. Now it is clear that (L) ~▫L((P₀(L,P) & L) ! P) and (M) ▫M((P₀(L,P) & L) ! P) are not explicitly contradictory. Someone might argue that (L) and (M) are implicitly contradictory. If logical possibility entails metaphysical possibility, then one gets the contradictory of (M); and then (L) and (M) are implicitly contradictory. Nevertheless, the dispositionalist has no motivation for accepting the premise that logical possibility entails metaphysical possibility. One might argue that we should expect a clear explanation of what metaphysical necessity is, since Bird’s account relies on it. This might be correct. However, it is not our aim in this chapter to defend Bird’s view but rather to show the consequences of his view for our discussion. How though could DE reject the ‘logically necessary’ reading of the box in ((P₀(L,P) & L) ! P) since we argued in Section 2 that that reading follows from the definition of determinism we adopt, which is not an unconventional one? It seems the trick would be to revise the definition of determinism so that it doesn’t involve logical necessity either but only metaphysical necessity, thus: Laws L are DE-deterministic iff for any P that L covers and any P₀(L,P) that is ‘appropriate’ input to L for P and any metaphysically possible worlds w, w’ in which L, if w and w’ agree on P₀(L,P), they agree on whether P obtains. This may indeed be a reasonable move for the DE advocate to make given the view that the only modalities that should play a role in these discussions about nature and its laws and possibilities are metaphysical ones. The second point concerns the question of free will. Lewis’s view gives motivation for rejecting one of the premises of the consequence argument, namely, the premise

³⁷ Beebee (2000, 578).

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that the laws of nature are not up to anyone since laws of nature supervene on the facts and some facts may be up to agents. On the other hand, it seems that those sympathetic to DE should accept this premise because they should, it seems, accept not only rule alpha but the rule α’: (α’)

▫MP ‘ ■P.

To see why, for DE, the laws of nature are not up to us, then, remember that for the dispositionalist the laws of nature are metaphysically necessary. Consequently, L is also metaphysically necessary. That is, 1.

▫ML.

Given rule α,’ from 1 we can derive 2.

■L.

So, it does not really matter in this case how we interpret ‘■.’ If rule α’ is valid, then proponents of DE should accept the premise that the laws of nature are not up to us.

5. Cartwright and Pemberton on Capacities and Arrangements Following the language of Cartwright, we call the kinds of powers that Cartwright and Pemberton defend ‘capacities.’³⁸ They do not use the term ‘manifestation’ since it would be ambiguous in their ontology. Capacities have a canonical way of acting, which is to be distinguished from what happens when they act. For each capacity, there is a prescribed set of ways in which it can act. When the capacity ‘gravity’ acts, it pulls, no matter what happens to the object on which it pulls. What actually happens depends on what other powers gravity cooperates with in the circumstances and what the arrangements are. When the arrangements are right, the activities of the powers give rise to regular behaviors, as in the orbits of the planets around the sun, or the browning of bread in a toaster, or the expulsion of magnetic fields in a superconductor. These kinds of arrangements are what Cartwright called ‘nomological machines’ and more recently are commonly called ‘mechanisms.’³⁹ Whether there are contingencies in nature then depends on whether all arrangements that occur in nature are like nomological machines or mechanisms, where it has been supposed that a single kind of behavior is fixed, or whether the outcomes can sometimes be open, and if so, how this is possible on a capacities account of laws. To begin with, we must be careful how we think of activities. The conditional account of dispositions and powers has it that for each disposition D there is a (possibly empty) set of stimuli S(D) and an outcome M(D) such that D obtains just in ³⁸ Cartwright (1989).

³⁹ Cartwright (1989).

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case if some s(D) ε S(D) occurs, then M(D), where s and M represent occurent features. Still in the grip of this account, we can slip into thinking of the activity as an occurent feature like s(D) and M(D). This makes for puzzles when powers act in consort. The outcomes of each power separately then seem to be pictured as ‘really there’ as outcomes, though it seems they are often invisible. The visible, or occurent, outcome is the result of the powers acting jointly. The model here is bricks in a wall. Each brick is really there and so too is the wall. Some real cases can be fitted into this model, for instance where the outcomes can be represented with numbers that simply add up. We have the power to put $10 into the piggybank and you do too. When we all act, the total outcome is $20 sitting there in the piggybank, $20 that is genuinely made up of our $10 and your $10. Perhaps it is not even too much of a stretch to fit forces into this model. When the gravitational capacity associated with mass M acts, it produces a force GMm/r² on another mass m located r from it; the Coulombic capacity associated with charge Q produces a force εQq/r² on a charge q located r from it. When both act together, they add vectorially. Perhaps we could without too much stretch say that all three forces are really there and in the same sense, as with the bricks and the wall. But, as Cartwright has argued, this is a poor model for other capacities acting in consort, like the capacities associated with parts of a circuit—conductors, resistors, impedances—producing a total current.⁴⁰ Nature may assign each capacity its own role, a role that it has qua the capacity that it is; and nature may fix what happens when capacities act in consort in given circumstances. But nature need not do this via a simple model where each capacity separately produces its own canonical effect and what results overall just is all these separate effects piled up together. It is because they want to avoid any suggestion of this picture that Cartwright and Pemberton abandoned their former language of capacities, contributions, and rules of combination that determine outcomes in favor of capacities, exercisings or actings, and outcomes, following Peter Machamer, Lindley Darden, and Carl Craver’s emphasis on activities.⁴¹ A capacity acts in a canonical way, which is represented in various different ways in scientific theories in different domains, and when capacities act in consort in a particular arrangement, an outcome occurs. For CartwrightPemberton capacities, there is indeed a difference between the obtaining of a power and its exercise, as there should be, in defense against Hume, who couldn’t keep these two distinct things in his ontology. But that does not make the exercising yet another occurent feature of the same kind as the resultant outcome. There is a second good reason for avoiding the language of contributions and how they combine. That makes it sound as if the capacity could act outside of any situation and the contribution is just what happens then. But capacities always act in some situation or other. We must not confuse the abstract description we give of a

⁴⁰ Cartwright (2009).

⁴¹ Machamer et al. (2000).

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capacity, which allows us to figure out what will happen in various real situations, with a description of what it does in some strange situation-less Platonic heaven. Perhaps we are sometimes led into this conflation by our conceptual model of the ideal experiment in which the capacity acts ‘entirely on its own,’ from which we sometimes read some canonical expression that we then use in making predictions about other circumstances, in accord with rules we have worked out about how to do this. But it is important to keep in mind that these idealized models picture concrete arrangements located in space and time, albeit ones that might never really occur. We emphasize this to underline the lesson that Cartwright and Pemberton stress: Arrangements matter.⁴² We may imagine two masses, M and m, located close together, m at r away from M, far away from anything else and also devoid of any other features, like charge, that are associated with a capacity to produce forces. Mass m would then experience a force very near to GMm/r², which is just the canonical description we give of the capacity of gravitational attraction in order to compute by our rule of vector addition the force exerted on a massive object in more complex arrangements. That however is an arrangement, albeit one very special one that we have discovered gives us a convenient way to represent the capacity of gravity for use in studying other arrangements. It takes a combination of a capacity with its peculiar nature (that in the case of gravitational attraction we represent by GMm/r²) and a given arrangement (like two objects both charged and massive being located close together and far away from all other objects) to fix what happens. This becomes important when we try to identify sources of contingency. What kinds of general facts are there that might get labeled ‘laws of nature’ on a capacities account? Three, it seems. First, what the nature of a power is. It is in the nature of the power of gravity to attract with a fixed strength. We represent this with the concept ‘the force of gravity’ and represent the strength of gravity associated with a body of mass M on m located r away by GM/r². Second, depending on one’s metaphysics of properties, laws should include facts either about what powers cooccur or what properties bring with them what powers. For example, mass brings the power of gravity with it; or, if properties are just to be collections of powers, we could label as a law of nature the general fact that the power to attract gravitationally comes with the power to resist acceleration by a force. Third, laws should include general facts about what happens when powers act, either singly or in consort, in various arrangements. For instance (supposing that resultant forces and not just motions are really there), when the power of gravity vested in M acts in a situation where m is located r away and no other sources of force on m are present, then M exerts a force GMm/r² on m. Or, in a situation where two powers we represent as forces act together on a body at a given point, then the body experiences at that point a force which is given by the vector sum of the canonical representations of the two powers.

⁴² Cartwright and Pemberton (2013).

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With this in hand, we can look to sources of contingency. Begin with permissiveness. On a capacity account of laws, permissiveness can arise along three axes: in the nature of the capacities themselves; in the rules of combination when capacities act together; and in the effects of arrangement on what happens, though the last two will merge except in special cases. The nature of a capacity. We can think of capacities as having three different possible modes of acting. First, there is one and only one mode of action for the capacity. No permissiveness here. Second, there is a set of available ways of acting and a probability over these. The capacity must act in accord with these probabilities. Versions of the propensity theory of quantum probabilities fit here. Third, the capacity may have an available set of modes of acting but no constraints on how often it acts in which ways even in the long run. Both these last two can lead to contingency about what happens when the capacity acts in specific arrangements. Rules of combination. The familiar rule of vector addition fixes a single force that results in arrangements where two sources of force act together. But we can imagine permissive rules that allow a range of outcomes, either with or without a probability over them. Then what results would be contingent in the permissiveness sense. The effects of the arrangement. Of course the effects of the arrangement are already there in the rules of combination. But we hive this off as a different source of contingency to deal with cases where the rules of combination do not cover all aspects of a situation that are relevant to what happens. We are thinking here of cases where experience shows that a given arrangement gives rise to some constraints or other on joint outcomes but there are no known rules of combination to explain this.⁴³ There is a general tendency in cases like this to think that the description of the situation is not detailed enough; when the details are filled in appropriately, there will be a general rule of combination to cover the case. That may—or may not—be so. The point is that there is nothing in the very notion of laws as facts about capacities and how they act in arrangements that precludes this source of contingency. Consider extent next. One may argue, as some do, that there is nothing but powers and their activities, in which case everything that happens must be the result of this. But just as with the other views of laws, this is an add-on to the basic account of what laws are for a capacities account and probably for most other powers accounts. Nothing about powers in themselves says they must rule everywhere and everything. As to reliability, the situation seems different. There seems no space in the capacity account to allow that things could happen within the domain of laws about capacities and about their joint outcomes that the powers-cum-arrangements do not allow. There seems to be no wiggle room on this account to allow that the laws of nature (which recall are about the natures of powers, what properties correlate with them, ⁴³ One might argue that where outcomes are constrained, there must be a rule of combination, albeit a very local one. That’s fine. We include this as a separate category to ensure attention is not focused entirely on well-established general rules of combination.

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and how they are to act in various arrangements) could be as they are and yet for something about which they speak to occur contrary to what they say. Or at least we have not identified any such wiggle room. Free will. This does not imply however that agents could not have acted otherwise than as they did. After all, the laws could be permissive. Or the actions of agents could be outside their domain. If though we insist on the analogue of determinism and universal domain for capacities, then it seems that on a capacities account an agent could not, consistent with those being the correct laws, have done otherwise than what she did do. There is an intermediate position even here. The laws for agents could be permissive consistent with those for non-agents being impermissive so long as only non-agents are involved. In that case, when agents and non-agents act together in certain arrangements, multiple outcomes could be available, including both, for example, that the agent raises her hand and that she does not. There will of course be trouble for this last alternative if it turns out that agents are just special arrangements of non-agents.

6. Conclusion and an Observation The observation is about our discussion of free will. For many, what we have discussed under this label is not only very cursory but also has little, even nothing, to do with free will because we have not touched on the ‘will’ part. Perhaps an agent could do differently from what she does but (and now the very form of this question itself is part of the serious enquiry) something like: ‘Can she do so because she wills it?’ Or, ‘Can she cause it to happen?’ After all, establishing that A’s actions could have been otherwise is a long way from showing that A is the author of her actions. Conversely, one venerable Christian tradition⁴⁴ along with some modern libertarian thought⁴⁵ argues that being the author of one’s actions does not imply that one could have done otherwise. Perhaps authorship is where attention should be in the contemporary debate anyway and not, as much seems to be, on the compatibility of free will with determinism since it has been a long time since our best science has supposed that the laws of nature are all deterministic. At least we hope to have clarified that even if laws govern and in some sense ‘make things happen,’ there is nothing in the very nature of law in any of the senses surveyed that implies that things couldn’t happen other than the way they do consistent with the laws staying the same, nor even that probabilities need be fixed. Laws may be universal in extent and yet totally impermissive, and one may—or may not—have good independent arguments for these add-ons; but in all senses of ‘laws’ surveyed that is just what these are: add-ons. ⁴⁴ Augustine (1993).

⁴⁵ Frankfurt (1969) and Mawson (2011).

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Conclusion: There are two surprises from this work, counter to our starting hypotheses. First, there are a number of different forms of contingency that are worth distinguishing and, contrary to initial expectations, contingency is no more readily admissible in any of these senses on a capacities (i.e. Cartwright and Pemberton powers) account of laws than on those that take laws as strong unifying regularities (BSA), as relations among universals, or as facts about dispositions of the Alexander Bird style (or as the metaphysically necessary facts about regularities that follow from these). All these equally can, but need not, allow laws to be both permissive and limited in extent. The second surprise is reliability. We use this label to pick out a view easy to say in plain English but hard to make precise, that the laws of nature may remain the laws they are, the correct laws, and yet be ‘violated’ or broken in their own domain. Violation—unreliability in our terms—fares badly on all accounts, except surprisingly, a David Lewis style best systems account, supposing we are willing to make an adjustment either to the notion of violation or to the BSA itself, where the adjustments rely heavily on a notion of ‘almost true.’ Under the soft determinism wiggle, though it is dressed up in the possibly impressive-looking quasi-formal language of possible worlds, the final verdict is that the correct laws are never violated in the strong sense. If something seemingly untoward happens (e.g. God intervenes), this can be a violation of some ‘almost true’ laws that prohibit it but not of the correct laws. The laws-as-templates wiggle gives up on the precise truth of the correct law claims: the regular associations that constitute the laws do not really hold; they only ‘almost’ hold. This is surprising in the context of much current discussion of ‘compatibilism’ in metaphysics—Is an interventionist God/free will compatible with deterministic law?—which seems to suppose the BSA. If we are right that reliability is unavoidable on the other accounts but could perhaps fail on the BSA, then this literature is focused on the easiest case for avoiding reliability.⁴⁶

⁴⁶ Thanks to the editors for helpful comments and thanks to Keith Ward for pressing issues of the extent of permissiveness. Cartwright’s research for this was supported in part by the Templeton-funded LSE/ UCSD project, God’s Order, Man’s Order and the Order of Nature and in part by the Durham project Knowledge for Use (K4U), which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 667526 K4U). The content reflects only the author’s view and the ERC is not responsible for any use that may be made of the information it contains. Merlussi’s research for this was supported by the CAPES Foundation.

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Index Aquinas, St. Thomas 1–2, 28–9, 32, 82, 84, 207, 212 and Descartes 21 and the hierarchy of being 23–4 and laws of nature 26–7 and teleology 22–3 Aristotle 24, 32–3, 35–7, 39–41, 69–71, 72–3 Armogathe, Jean-Robert 28–9 Armstrong, D. M. 5–7, 9, 127, 156–7, 218–19, 232–3 Azzouni, Jody 190–1 Bacon, Francis 42 n. 3, 62–3, 72–4, 79, 207 and forms 64–9 Basso, Sebastian 21, 33–4, 36–41 Beebee, Helen 123, 125, 227, 237 Beeckman, Isaac 19–21, 32, 36 n. 87, 40–1 Berkeley, George 6–9, 12, 81, 95–7 best system account 7–8, 14–16, 139–60, 166, 171–2, 175–6, 225–32 Bird, Alexander 160–1, 169, 222, 234–8, 244 boundary conditions 15, 168, 222, 227–8, 232, 237 Boyle, Robert 3, 43 n. 4, 170, 208 Briggs, Rachel 149 Callender, Craig 14, 15 n. 54, 143 n. 8, 146–9, 225 Carnap, Rudolf 10–11, 13, 148 n. 20 Cartwright, Nancy 7 n. 25, 10 n. 37, 13–14, 42, 206, 222, 239–41 causation 28, 135–7, 143, 183–6, 191–4, 213–14, 224 see also regularity; Hume, David; powers; necessity, nomic ceteris paribus clauses 14, 207–11, 217–20, 236 Cohen, Jonathan 14, 15 n. 54, 143 n. 8, 146–9, 225 cohesion 53, 193–4 concurrentism 19 n. 8, 31 contingency 97, 154, 211–12, 214, 221–7, 230, 235, 241–4 Copernicus, Nicolaus 46 n. 24, 152 n. 30 counterfactuals 8, 157, 169–71, 174–5, 216, 218, 232–4 see also necessity, nomic Cudworth, Ralph 4–5, 62 Curley, Edwin 69 n. 42, 75 n. 58, 76–7 deductive-nomological model 11–12 Dennett, Daniel 191

Descartes, René 26–7, 31–41, 42–55, 59–64, 67–9, 74–9, 80–6, 144–5 and divine immutability 4, 19, 29–30, 46–7, 76, 86–8, 98 and laws of motion 3, 20, 29–31, 67–8 and mathematics 2–3, 20–1 and occasionalism 5, 62, 78, 91 determinism 177–8, 221–3, 227–32, 237–8, 243–4 dispositional essentialism 112, 146, 156, 158–9, 234–9 dispositions 9, 29–31, 38, 66–70, 74–6, 88–9, 176–7, 207, 210–20 see also powers Dretske, Fred 5–7, 232, 233 Duhem, Pierre 12 Earman, John 14–15, 226–7 Einstein, Albert 144 Ellis, Brian 9, 79 free will 223, 231, 238, 243–4 and God 43, 105 Galilei, Galileo 30, 46 n. 24 Garber, Daniel 20 n. 13, 89 n. 43 Gassendi, Pierre 28 n. 49, 29 n. 56, 64 n. 12 Giere, Ronald 149 Goodman, Nelson 11, 147 gravity 7, 85, 103–4, 143, 191, 196–7, 214–16, 239–41 Newton on 52–3, 64 n. 16, 78 n. 73, 103–4, 143 Halpin, John 146–51, 55 Hattab, Helen 3 n. 9, 5 n. 16, 46 n. 23 Haugeland, John 190–1 Hempel, Carl 11–12, 15 Hobbes, Thomas 81 n. 2 Hume, David 6–7, 101–2, 107, 122–4, 206, 212, 225–8, 250 Humeanism see best system account; Lewis, David initial conditions 15, 126, 150 n. 29, 158–78, 222, 227–8, 232, 237 invariance 158–77, 179 Kant, Immanuel 108–21 Kepler, Johannes 19–21, 152 n. 30, 162

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INDEX

Kitcher, Philip 111 Kratzer, Angelika 135 Kuhn, Thomas 151, 153 n. 31, 154 Lange, Marc 127, 144, 170 n. 33, 173, 186 n. 12 Lavoisier, Antoine 141–2, 144, 152–3, 155 Leibniz, G. W. von 43 n. 4, 81, 97–101, 104–7, 142 Lewis, David 4, 132, 169, 181, 187 see also best system account Lipton, Peter 207, 217 Locke, John 64 Loewer, Barry 5 n. 19, 6 n. 23, 8, 126 n. 13, 186 logical empiricism 10–17, 225 logical positivism see logical empiricism Lowe, E. Jonathan 207, 218–19 McLean, Ian 18 Malebranche, Nicolas 81, 85–6, 91–100 Maudlin, Tim 122, 124–7, 131, 137 mechanism 14, 67–8, 97, 100–1, 192, 239 Messina, James 110, 111 n. 11, 112 n. 16, 113 Mill, John Stuart 207, 215–17 miracles 14, 26, 100, 104, 206 divergence miracle 229–30, 232 mirror argument 122–5, 131, 136–7 Nadler, Steven 76 n. 63 natural kinds 3, 68, 70–3, 139, 147, 218–19 natural philosophy 6, 16–48, 53–69, 80–2, 96–105 necessity 30, 80–2, 86, 105–7, 159, 176, 214, 232–4 and causation 6 n. 24, 94–102 logical 5, 237–8 nomic 108–14, 119–20, 146–8, 150, 155–7 see also determinism, powers, occasionalism Neoplatonism 23, 32–36, 39–40, 152 n. 30 Neurath, Otto 10 n. 37, 13–14 Newton, Isaac 5, 7, 43–5, 52–60, 81, 102–7, 143, 173, 207 see also gravity, Newton on occasionalism 36, 91–100 see also Descartes, René; occasionalism Oppenheim, Paul 11–12 Pemberton, John 222–3, 239–41, 244 Piccolomini, Francesco 21, 30, 33–7, 40 Plato 21, 29, 33, 38, 78, 234, 241 see also Neoplatonism Poincaré, Henri 12, 178

powers 1–4, 16, 70–2, 79, 80–4, 98–104, 106–7, 239–42 see also dispositions properties 14, 36, 52–3, 56–7, 73–4, 85, 130, 241 categorical/quiddities 9, 146, 150, 157, 210–13 natural 139–56, 181–201 Psillos, Stathis 7 n. 26, 10 n. 39, 81 n. 2, 101 n. 120, 169 Putnam, Hilary 14, 148 quiddities 9 Ramsey, Frank 14–15, 147, 225 reducibility 181–204 regularity 20, 40, 80, 93–6, 106–7, 112–19, 128–9, 161–6, 209–10 Humean 101, 123, 221–3, 225–7, 230 see also best system account relations 42, 78, 115–16, 120, 127, 159–60, 162, 172–4 and necessitation 146, 150, 156, 174 among universals 5–6, 221, 232–4, 244 Roberts, John 14–15, 123, 126–7, 130–2, 135–7 Schmaltz, Tad 46 n. 23, 49 n. 36, 71 n. 47 Schrenk, Markus 210–12 Scientific Revolution 79 n. 75, 221 Sehon, Scott 227–8, 230, 237–8 semantics 123, 132–8, 170 simplicity 140–7, 151–6, 159, 166, 175 divine 47, 76, 87, 91–2, 96–9 Spinoza, Baruch 9, 62–3, 68–79 Stoicism 30, 33, 37–40 Strevens, Michael 184, 188, 192–4, 203 Suarez, Francisco 1–2, 29, 82 supervenience 16, 124, 130, 139, 146, 150, 225 symmetry 15 n. 52, 159, 163, 177 systematicity 109–19 theology 3, 5, 6, 9, 10, 22, 28–30, 207 Tooley, Michael 5–7, 122 n. 2, 127, 232, 233 universals 5–7, 122, 159, 176, 221, 232–6, 244 Whewell, William 187–8 Wigner, Eugene 159, 161–8 Wilson, Mark 168 Wimsatt, William 230 Woodward, James 15 n. 52, 159 n. 11, 166 n. 26 Zabarella, Giacomo 21, 30, 33, 35, 37, 40