Oxford Handbook for Isaac Newton Chris Smeenk Western University Eric Schliesser University of Amsterdam

Newton's Regulae Philosophandi Zvi Biener Abstract. Newton's Regulae philosophandi - the rules for reasoning in natural philosophy - are maxims of causal reasoning and induction. This chapter reviews their significance for Newton's method of inquiry, as well as their application to particular propositions within the Principia. Two main claims emerge. First, the rules are not only interrelated, they defend various facets of the same core idea: that nature is simple and orderly by divine decree, and that, consequently, human beings can be justified in inferring universal causes from limited phenomena, if only fallibly. Second, the rules make substantive ontological assumptions on which Newton's argument in the Principia relies. Keywords: Newton, method, universality, transduction, induction, fallibalism, simplicity Newton's Regulae philosophandi - the rules for reasoning in natural philosophy - are maxims of causal reasoning and induction. They appeared in some form in all three editions of the Principia (1687, 1713, 1726), where they opened Book III of the work - the book in which Newton "come[s] down to Physics" from the mathematical peaks of Books I and II and considers natural causes and their scope (Principia, 588).1 Their location in Book III mirrors the location of the "Axioms, or Laws of Motions" of Book I, and for good reason. If for Newton the "main Business of natural Philosophy" was to "argue from the Phaenomena without feigning Hypotheses, and to deduce Causes from Effects," the rules provided the principles for doing so (Opticks, 369). In the Principia, they were used to deduce the work's controversial new cause: universal gravitation. As universal gravitation became increasingly accepted, the rules themselves struck root and became philosophical orthodoxy. Leonard Euler used them to justify his concept of force, William Whewell sought to subsume them under his comprehensive inductivism, and Charles Darwin structured the Origin of Species to meet their demands (Harman 1983, Butts 1993, Hodge 1977). For some, their authority was even a mark of self-evidence. Thomas Reid declared the rules to be "maxims of common sense. . . practised every day in common life." For him they were so central 1I refer to the Principia's editions as E1, E2, and E3, and to the rules as R1, R2, etc. I refer to propositions by book, proposition, and corollary; e.g., III.6.c2 is corollary 2 of proposition 6 of book III. Principia quotations are from Newton (1999). Opticks quotations are from Newton (1952). Correspondence quotations are from Newton (1959–1977). 2 Newton's Regulae Philosophandi to the enterprise of knowledge that "he who philosophizes by other rules. . . mistakes his aim" (Reid [1764] 1997, 12). But the commonsensical nature of the rules belies a complex history. They were called "Hypotheses" in E1 and only relabeled "Regulae Philosophandi" in E2 and E3.2 In E2, Newton added a new rule, R3, which was kin to but substantially different from a hypothesis he chose to remove, H3. In E3, he added yet another rule, R4, and expanded the explanatory text of R3 to address debates concerning the status of gravity as an essential property of matter. He also rephrased the remaining rules in a distinctly cautious, 'epistemic' tone. Newton even entertained a fifth rule (concerning the epistemic value of introspection), but never published it. These are not the only changes. A variety of finer ones - some discussed below - can be traced through Newton's manuscripts and his annotated and interleaved copies of the Principia over a period of nearly forty years. Their scope, the years separating them, and the clear implication that Newton did not consider the rules to have reached a complete form led I. B. Cohen to note that "not only is it anachronous to lump together [the rules, as well as other texts], as if they all represented Newton's state of mind at one and the same time: to do so is also to deny the dynamic quality of a creative mind by assuming that there were no changes on fundamental questions during a span of some forty years" (Cohen 1971, ix). This chapter will provide an overview of the rules and their history. Although I consider various issues, two claims will emerge. First, the rules are not only interrelated, they defend various facets of the same core idea: that nature is simple and orderly by divine decree, and that, consequently, human beings can be justified in inferring universal causes from limited phenomena, if only fallibly. Second, the rules make substantive ontological assumptions on which Newton's argument in the Principia relies. Although the chapter is more expository than argumentative, these claims go to challenge the idea that the rules are superfluous and merely formalize inferences that Newton was capable of making on entirely empirical grounds, as well as the idea that the aim of the rules is to express a sophisticated methodology Newton was the first to formulate. I call this a 'conservative' interpretation of the rules. I begin by discussing the overall purpose of the rules (§2), and then examine them in descending order (§3–§5). 1.1 Analysis, Synthesis, and the Purpose of the Rules Newton's extensive revisions to the rules were almost entirely driven by debates with Cartesians and Leibnizians. Their crux was already well expressed in the Principia's first review, by the Cartesian Pierre-Sylvain Régis: 2The relabelling occurred between 1694 and 1706, most likely in the 1700s. Although it is important, it does not indicate that Newton initially thought of the rules as mere hypotheses. Newton was following a convention of contemporary physico-mathematical works (like Huygens's Horologium Oscillatorium) according to which "hypotheses" are starting points of physical explanation. Conversely, the relabeling does not indicate that Newton came to think of the rules as unassailable. Rather, he thought of them as broadly reliable principles that could nevertheless admit of exception or inexactitude - rules of thumb, so to speak. Achinstein (2013) calls them "rules of strategy," Stein (1967) calls them "guiding principles." Newton even suggested that the rules were the results of induction, and so as fallible as induction (McGuire 1970, 70). Analysis, Synthesis, and the Purpose of the Rules 3 The work of M. Newton is a mechanics, the most perfect that one could imagine. . . [But] one cannot regard [his] demonstrations otherwise than as only mechanical; indeed, the author recognizes. . . that he has not considered their Principles as a Physicist, but as a mere Geometer. . . He confesses the same thing at the beginning of the third book, where he endeavors nevertheless to explain the System of the World. But it is only by hypotheses that are, most of them, arbitrary, and that, consequently, can serve as foundation only to a treatise of pure mechanics. (Journal des Sçavans, 2 August 1688, 153ff; in Koyré 1965, 155) Régis appealed to a common distinction between physica (or philosophia naturalis) and mechanica. The first was thought to provide causal knowledge of natural change, the second knowledge that was perhaps instrumentally useful, but did not capture natural change through its real causes.3 Régis's charge was that despite the Principia's commendable treatment of mathematics, it did not capture real causes. He singled out the beginning of book III - the location of Newton's Hypotheses and later Regulae - as the point at which Newton employed principles that were arbitrary, and so precluded the possibility of a genuine physica.4 This was a clear affront to Newton's conception of natural philosophy. To him, the "main business of natural philosophy" was to engage in what late scholastics called the regressus - a two-stage process of reasoning from effects to causes (the analytic or reductive stage) and then from causes back to effects (the synthetic or compositive stage).5 The process was aimed squarely at causal knowledge. In Newton's words: [I] natural philosophy. . . the method of analysis, ought ever to precede the method of composition. This analysis consists in making experiments and observations, and in drawing general conclusions from them by induction. . . By this way of analysis we may proceed . . . from effects to their causes, and from particular causes to more general ones . . . [T]he synthesis consists in assuming the causes discovered. . . and by them explaining the phenomena proceeding from them, and proving the explanations. (Opticks, 404–5) The rules played an essential role in this two-stage process, at least in the Principia. Their role was similar to that ascribed by some late scholastics to a third stage of investigation, the negotiatio intellectus (or contemplatio mentalis). The negotiatio separated analysis from synthesis and was thought to consist of mental operations that, first, winnowed down the set of causes discovered by analysis to only true causes and, second, allowed the natural philosopher to grasp how these causes could be applied universally despite being inferred from limited phenomena. The negotiatio was essential for the regressus, as it determined which causes were taken as fundamental in the synthetic stage (Randall 1940, 192–201). To my knowledge, Newton never used the term negotiatio. However, the idea is useful for us, since it provides a rather accurate description of the function of the 3Even before Newton, this blunt distinction was debated and this debate was crucial for the rise of mathematical physics (Mancuso 1997, Dear 1995). 4One of the principles Régis considered arbitrary was universal gravitation itself. Leibniz and his followers also held that universal gravitation was not metaphysically well founded. For the Leibniz's criticisms, see Koyré (1965, 115–38), Bertoloni Meli (1993), Vailati (1997). Particularly clear is Leibniz's "Against Barbaric Physics" (Leibniz 1989, 312–20). This is not to say that all objections to the Principia were of a broad methodological and metaphysical nature. There were many mathematical and evidentiary reasons to seek corrections to Newton's theory and to maintain agnosticism pending further evidence. 5For Newton's familiarity with the Scholastic textbook tradition, see Ducheyne (2012, Ch. 1). For the regressus, see Poppi (2004), Wallace (1995). 4 Newton's Regulae Philosophandi rules. 6 To begin with, the rules's placement at the beginning of Book III indicates they are not precepts regarding the practice of natural philosophy in general. If this was their purpose, we would expect to find them at the start of the entire Principia. Instead, their placement indicates that Newton was aware that principles of thought were required to negotiate the transition from an abstract treatment of forces to causal reasoning about the actual world.7 The rules are principles for, first, winnowing down the multiplicity of possible causes of celestial phenomena to a single true cause - gravity - and, second, for allowing gravity to be taken universally, despite being inferred from a limited range of phenomena. Like the negotiatio, the rules are guidelines for thinking, not for direct empirical investigation. And like the negotiatio, they provide the necessary link between the Principia's analysis (i.e., the determination of the various instances of force involved in basic celestial phenomena) and synthesis (i.e., taking those instances as manifestations of the same cause - gravity - and using gravity to explain additional phenomena). Newton's repeated changes to the rules were meant to clarify how they can execute this function, and thus to answer the charge that the Principia had not, perhaps could not, provide genuine causal knowledge.8 Let's see how the rules do this. 1.2 Rule 4: Provisionalism and Induction Rule 4 In experimental philosophy, propositions gathered ( collectae) from phenomena by induction should be considered either exactly or very nearly ( aut accurate aut quamproxime) true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact ( accuratiores) or liable to exceptions. This rule should be followed so that arguments based on induction may not be nullified by hypotheses. (Principia, 796). R4 was added to the Principia in E3 (1726).9 Although it was largely ignored in the subsequent century (more on this below), it is nowadays the most discussed of Newton's rules. To understand it, we begin with Newton's long-standing opposition to "contrary hypotheses." It was first expressed in his public debates about the nature of light with Robert Hooke, Gaston Pardies, and Christiaan Huygens. In 1672, Newton claimed to have established with certainty, on the basis of experiments, that white light was composed of rays of different "refrangibility," with each degree of refrangibility corresponding to a different color. The claim elicited fierce objections. First, Newton's opponents held that since "refrangibility" was not a mechanical quality, its use in physical explanations - and thus Newton's entire account - was suspect. Second, they held that although Newton's account was consistent 6Although Newton did not use the term, education in Cambridge included several treatments of Aristotelian logic that appealed to the concept, if not the word (Sgarbi 2013, 35–9). In general, in the latter half of the seventeenth century the function of the negotiatio was often folded into an inductive component of analysis (Jardine 1976[28ff.]; Dear 1995). This is how Newton conceptualized it. 7I thank Mary Domski for this point. 8There are two important corollaries to this way of contextualizing the rules. The first is that they do not capture Newton's method as a whole, but only a single, albeit crucial, element of it. The second follows from the idea that the negotiatio "often appealed to information or explanatory principles beyond the original subject matter" (Barker and Goldstein 2001). This characterization belies the claim that the rules merely formalize reasoning that Newton could justify on purely empirical grounds; see Di Fate (2011), Spencer (2004), and Harper (2011). 9Portions of this section are also discussed in Biener (forthcoming). Rule 4: Provisionalism and Induction 5 with the phenomena, it was only one among many possible "hypotheses." To establish this account with certainty, Newton would have to eliminate all alternatives, and this was impossible. In general, the threat of what we now call 'underdetermination' dominated natural philosophy in the latter half of the seventeenth century. Newton's opponents, like many of their contemporaries, believed that humans were in an essentially restricted epistemic position: they could offer competing hypotheses to save the phenomena, but could never 'see-through' the phenomena to determine which hypotheses were true.10 At best, they could reject hypotheses that were empirically inconsistent or did not abide by some privileged non-empirical considerations (e.g., an exclusive appeal to mechanical qualities). But no more discriminating procedure for theory selection was possible.11 Newton agreed that an eliminative method for theory selection was inadequate (precisely because "numerous hypotheses may always be devised"), but suggested that its failure ought to focus attention on the context of discovery and the process of theory creation (Correspondence, I, 164). He wrote: I cannot think it effectuall for determining truth to examin the severall ways by wch Phaenomena may be explained, unless where there can be a perfect enumeration of all those ways. . . [T]he proper Method for inquiring after the properties of things is to deduce them from Experiments. . . [T]he Theory wch I propounded was evinced to me, not by inferring tis thus because not otherwise, that is not by deducing it onely from a confutation of contrary suppositions, but by deriving it from Experiments concluding positively & directly. The way therefore to examin it is by considering whether the experiments wch I propound do prove those parts of the Theory to wch they are applyed, or by prosecuting other experiments wch the Theory may suggest for its examination. (Correspondence, I, 209, original emphasis) For Newton, a proper theory is discovered by "deduction" from phenomena; that is, phenomena are used "positively & directly" to entail the theory that explains them, not only negatively to disqualify theories that are inconsistent with them. The logical, evidentiary tie that such a deduction establishes between phenomena and theory renders the theory immune to alternatives that are not themselves grounded in the phenomena or, worse, the mere possibility of alternatives. Such alternatives, as intuitively compelling as they may be, do not directly pertain to the established tie between phenomena and a deduced theory, and so cannot sever it. R4's purpose is to rebuff "contrary hypotheses" of this sort. Already in the 1670s, Newton was clear in his opposition to them.12 Yet R4 does not merely reiterate Newton's 1670s methodological stance. In the early debates, Newton had a zealot's confidence that natural philosophical claims can 10A locus classicus is Descartes's Principles of Philosophy, IV, §204. 11The methodological stances of Hooke, Pardies, and Huygens were actually subtly different. For example, Huygens held that underdetermination was a theoretical but mostly unreal threat. He believed that consistency with phenomena was so hard to achieve that consistent propositions have "a probability nearing truth" (Huygens 1690b, 125). See Shapiro (1989) and Zemplén and Demeter (2010). 12Newton's replacement of Hypotheses with Regulae Philosophandi was likely meant to suggest that the principles of Book III were not hypothetical in this way. It is also possible, however, that their replacement was a response to the 1701 Latin publication of Descartes's Regulae ad directionem ingenii (in Opuscula posthuma, physica et mathematica, Amsterdam). Perhaps Newton modified his Principia to jab once again at Descartes. 6 Newton's Regulae Philosophandi be known with mathematical certainty. By the 1710s, when R4 was articulated, his views had become considerably more nuanced and cautious.13 They resulted from his development, in the process of composing the Principia, of what I.B. Cohen called "the Newtonian style" See (Cohen 1982).14 Briefly, on the Newtonian Style real-world motions are approached by a series of increasingly accurate approximations. These are underwritten by "quamproxime" propositions - propositions relating motions to forces in which the conclusions are approximately true iff the premises are approximately true (G. E. Smith 2002b). An increasingly accurate characterization of motions and forces is achieved by specifying the conditions under which an initial approximation holds exactly and then taking systematic deviations from the exact conditions as physically significant; i.e., taking them to have yet-unspecified physical sources and beginning a search for such sources. When these are found and their effects characterized by a new approximative model, the conditions under which it holds exactly become fodder for the next search for sources of deviation, and so on, iteratively. There is no guarantee, however, that the iterative process will always succeed. If at any point the process is hopelessly frustrated, the assumptions on which it is based - concerning the sources and nature of forces, their strengths, etc. - may be rejected. Even the most fundamental assumptions of Newton's system - e.g., the Laws of Motion on which the entire investigative framework rests - may be rejected.15 In that sense, all claims in Newton's system are provisional; they can be accepted as long as the iterative process continues successfully, but may be rejected in light of future research. R4 captures the main elements of this style. First, the style's provisionalism is reflected in R4's concern with the attitude of investigators towards the propositions they entertain (propositions should be considered true, and so taken as holding exactly), not the objective truth of those propositions. Second, the style's reliance on approximative techniques explains R4's instruction that some propositions should be taken only as very nearly ("quamproxime") true, not just true simpliciter. Finally, the style's provisionalism and reliance on approximations sanction the program for continuing research suggested by R4: in the course of future investigation, propositions ought to be made "either more exact or liable to exceptions." But there are also reasons to be cautious about reading the Newtonian style into R4. Newton's provisionalism, as well as his condemnation of contrary hypotheses, are compatible with reading the rule as a defense of induction from instances.16 On 13Even in the 1670s debates, Newton ultimately weakened his position, conceding that the certainty of physical theory was not identical to that of mathematics (Shapiro 1993, 12ff. Guicciardini 2009, 19ff.). In the 1670s debates, Newton also tended to run together inductive and deductive 'moves.' Later in life, he was more careful to separate them. 14Our understanding of the Newtonian style, too briefly sketched here, is due to the groundbreaking work of George Smith and William Harper. See, in particular, Harper and G. E. Smith (1995), G. E. Smith (2002a), Harper (2007), Harper (2011), and G. E. Smith (2014). 15This is precisely what happened in the transition from Newtonian to Einsteinian gravity. See G. E. Smith (2014) and Harper (2011, Ch. 10). The prospect of revising the laws illustrates clearly that although the laws open the 'mathematical' portion of the Principia, they are not purely mathematical propositions. They are open to overthrow by empirical finding. 16By induction from instances, I mean a process that "set[s] forth a certain number of cases and observe the resulting [properties], and then compare[s] them with one another in order that the universal proposition can then be known" (Jonn Wallis, Opera, I, 365; in Guicciardini (2009, 142)). Rule 4: Provisionalism and Induction 7 this deflationary reading, Newton's focus on the attitude of investigators indicates his recognition of the inherent risk involved in even the simplest inductive generalization. Moreover, his demand that some propositions not be taken as strictly true and that all propositions be subject to future correction merely indicates his recognition that inductions can be made stronger/weaker and more/less general by new findings. The primary evidence for this view is R4's use in the body of Principia, the circumstances of its initial formulation, and the various contexts in which Newton made statements similar to it. I rehearse this evidence at some length, since it highlights the interrelatedness of R4 and R3, one of the themes of this chapter. R4 is used only once in the Principia, in defense of proposition E3.III.5. That proposition (in all editions) argues that gravity is a mutual interaction, and that thus all planets and moons "gravitate toward one another;" for example, the sun gravitates towards the planets as the planets gravitate towards the sun, and planets gravitate towards their moons as the moons gravitate towards their planets. It relies on two assumptions. First, it requires that the third law of motion (the action/reaction law) be applied to the centripetal forces holding satellites in their orbits. It is the third law that turns these into mutual interactions. Second, the proposition requires identifying the bodies towards which these centripetal forces point as their sources. This assumption turns the forces into interactions between satellites and the central bodies around which they revolve. In 1713, Roger Cotes - the editor of E2 - pressed Newton on these assumptions. The crux of his objection was that while Newton had good grounds for applying the third law to contact action, he had no evidence how, or even that, the law applies to non-contact, gravitational action (Biener and Smeenk 2012, 119–24). To make the point vivid, Cotes asked Newton to entertain a hypothetical "invisible hand" - a non-gravitational, mechanical cause able to push a satellite towards its central body. If this hand were the cause of the satellite's acceleration, the third law would entail that it, not the central body, be subject to a reaction force. How did Newton know that the third law applied to the central body, not the hand? Cotes surely galled Newton by declaring that "'till this objection be cleared I would not undertake to answer one who should assert that You do Hypothesim fingere" (Correspondence, V. 392). Newton responded by dismissing Cotes's hypothetical cause and defending the third law. Some of his comments concerned the close connection between the first law (the law of inertia) and the third. Others concerned the methodological validity of generalizing the third law from contact to non-contact action; that is, the validity of inducing the law from known to unknown instances. It is here that Newton first clearly articulated the line of thought that resulted in the addition of R4 to E3 some thirteen years later. In a draft response, he brushed off Cotes's objection with: Experimental philosophy argues only from phenomena, draws general conclusions from the consent of phenomena, and looks upon the conclusion as general when the consent is general without exception, though the generality cannot be demonstrated a priori. . . [I]n experimental philosophy it's proper to distinguish propositions into principles, propositions, and Wallis's procedure concerns a mathematical context different from the one we are considering, but his terminology is strikingly similar to Newton's. 8 Newton's Regulae Philosophandi hypotheses, calling those propositions which are deduced from phenomena by proper arguments and made general by Induction (the best way of arguing in philosophy for a general proposition) and those hypotheses which are not deduced from phenomena by proper arguments. (Correspondence, V. 398–399) For Newton, the invisible hand - since not directly implied by the phenomena - was a mere hypothesis, to be dismissed for the reasons discussed above. The third law, in contrast, was gathered from phenomena by proper arguments (offered in the scholium to the Laws of Motion) and made general by induction. It was known to be true in well-understood, experimented-upon instances (contact forces), presumed by induction to be generally true, and so extended to less-understood instances (celestial centripetal forces). Newton knew this extension came with no guarantees - the law's generality "cannot be demonstrated a priori" - but he also believed Cotes did not provide good reason to doubt it. Newton's defense of the third law, in short, appealed to the validity of induction. His position, later captured by R4, was that the results of induction are presumptively true and trump the mere possibility (logically feasible, but not empirically supported) of contrary instances. The Newtonian style was not at issue here. Proposition III.5 does raise questions about mutual interactions that can be answered by means of the sophisticated style, but neither Newton nor Cotes saw these as the relevant points of discussion. Newton also made statements similar to R4 in other contexts. Take, for example, Query 31 of the 1717 Opticks: [Hypotheses] are not to be regarded in experimental Philosophy. And although the arguing from Experiments and Observations by Induction be no Demonstration of general Conclusions; yet it is the best way of arguing which the Nature of Things admits of, and may be looked upon as so much the stronger, by how much the Induction is more general. And if no Exception occur from Phenomena, the Conclusion may be pronounced generally. But if at any time afterwards any Exception shall occur from Experiment, it may then begin to be pronounced with such Exception as occur. (Opticks, 404) The context of the passage is revealing. Query 31 begins by asking "Have not. . . Bodies. . . certain Powers. . . by which they act at a distance[?]" It continues by listing a variety of physical, chemical, and biological phenomena that exemplify powers, and concludes that "thus Nature will be very conformable to her self. . . Performing all the great Motions of the heavenly Bodies. . . [and] almost all the small ones [by similar powers]" (Opticks, 375, 397, emphasis added). In other words, the query is a long inductive argument, and the passage above comments on it. Immediately after the passage, Newton also supplies a précis of his overall argument in the Opticks. But the Newtonian style is foreign to the Opticks, and, appropriately, the passage focuses on the generality and strength of induction from instances. The deflationary view of R4 is perhaps best supported by one of the rule's less apparent features, its close connection to R3, a rule that is certainly not concerned with the sophisticated approximative style of the Principia. In Newton's final letter regarding III.5, he instructed Cotes to add the famous Hypotheses non fingo passage to the General Scholium.17 It contains another echo of R4, but also an appeal to those qualities that are the subject of R3: 17At the time of the exchange, Newton could not make modifications directly to III.5, as that portion of E2 had already been printed. Rule 4: Provisionalism and Induction 9 In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies, and the laws of motion and the law of gravity have been found by this method. (Principia, 943) Newton's draft letter contained a variety of similar statements. In fact, he explicitly notes that the inductive generalizations of III.5 "holds good by the third Rule of philosophizing" (Correspondence, V 398, emphasis added). He repeatedly appealed to (forms of) R3 to explicate R4, and even noted that the R3's rejection was tantamount to "destroy[ing] all arguments taken from Phenomena by Induction," a phrase that mirrors R4's "This rule should be followed so that arguments based on induction may not be nullified" (Correspondence, V 398, Principia, 796).18 The connection between R3 and R4 was apparent to Newton's followers. For example, Henry Pemberton, the editor of the edition in which R4 first appeared, wrote that on R3 "is founded that method of arguing by induction, without which no progress could be made in natural philosophy," and added that "[Newton] farther inforces [the method of Induction] by this additional precept [R4], that whatever is collected from this induction, ought to be received, notwithstanding any conjectural hypothesis to the contrary, till such times as it shall be contradicted or limited by farther observations on nature" (Pemberton 1728, 25–6). To Pemberton, R4 was merely an extension of R3, not an additional precept regarding quamproxime reasoning or the inferential style of the Principia. Pemberton was a mediocre mathematician, however, and it is no surprise that he didn't arrive at more sophisticated interpretations. Still, one cannot blame him. Even an attentive reader of the Principia and Opticks could have easily juxtaposed R4's "considered. . . exactly or very nearly true" with the Opticks's "may be looked upon as so much the stronger, by how much the Induction is more general," and considered R4 to be a further reflection on induction from instances. Perhaps this is why R4 was mostly neglected in the century after Newton's death. Without a good deal of mathematically sophisticated Newtonian scholarship, the rule does indeed seem to contain, as Whewell averred, "little more than a general assertion of the authority of induction, accompanied by Newton's usual protest against hypotheses" (Whewell 1840, 452).19 We have seen, however, that even in Newton's mind, R4 was tightly bound with the nature of enumerative induction, particularly its fallibility and the consequent provisionalism of natural philosophy. While this fallibility is the focus of R4, in R3 Newton takes a more assertive, confident tone.20 Let's turn to R3, Newton's most 18In drafts to R4, Newton also claimed that without that rule "the arguments of inductions on which all experimental philosophy is founded could always be overthrown," a claims that mirrors his assertion, in the Principia, that R3 is "the foundation of all natural philosophy" (Koyré 1965, 269). I should also note that in an annotated copy of E1, Newton explicitly justified III.5.c2 "per Hypoth III." 19One wonders how R4 would have been received if John Keill, one of Newton's earliest expositors, had lived past 1721. In lectures delivered in 1700, he stressed that in Newtonian natural philosophy "it is necesary to make use of a more lax sort of Reasoning, and to exhibite Propositions that are not absolutely true, but nearly approaching to the Truth [ad veritatem quam proxime accendentes]. As, for example, when it is demonstrated that all the Vibrations of the same pendulum made in the small Arches of a Circle, are of equal Duration" (Keill 1702, 74, Keill 1720, 88). Keill made no mention of "induction" in this context. 20The contrast between R4's provisionalism and R3's confidence is discussed further in Biener and Schliesser (2017). 10 Newton's Regulae Philosophandi explicit statement about taking causes and propositions regarding them universally. 1.3 Rule 3: Induction and Universality Rule 3 Those qualities of bodies that cannot be intended and remitted and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally. [These include extension, hardness, impenetrability, mobility, inertia, and gravity.] R3 is the most historically and conceptually complex of Newton's rules. Like R4, it was used only once in the Principia. In proposition III.6.c2 of E2 and E3, it supported the claim that "all bodies universally" gravitate (literally: are heavy) in proportion to their quantities of matter (Principia, 809). Based on pendulum (and later free-fall) experiments, Newton determined that at equal distances from the earth, the weights of bodies made of any material (wood, gold, wheat, etc.) was directly proportional to their quantities of matter.21 Since this proportion could not be "intended or remitted" - i.e., could not be changed by material operations - and since it was "a quality of all bodies on which experiments can be performed," it followed by R3 that it could be generalized and "affirmed of all bodies universally" (Principia, 809). III.6 was the first proposition in which Newton asserted that weight is possessed by every body, regardless of size, spatial position or kind. The inference was intended to ward off the possibility of bodies that do not gravitate at all or gravitate differently, either the small aetherial particles many believed were responsible for gravitation, or bodies at planetary distances that were not directly accounted for by Newton's evidence. It was also intended to ward off the idea that gravity, which had been compared to magnetism by the likes of Robert Hooke and Gilles Personne de Roberval, did not act like magnetism: it affected all bodies, not a subset of them. Curiously, gravity's universality was justified differently in E1. That edition contained an "Hypothesis 3" (henceforth, H3) which read: Every body can be transformed into a body of any other kind and successively take on all the intermediate degrees of qualities. (Principia, 795)22 This "transmutation hypothesis" was replaced by the familiar R3 in E2 and E3. The change is less substantial than it first seems. Although H3 was abandoned, the argument it supported remained in E2 and E3, albeit in an ancillary role to the main inductive argument. Newton argued that if there were kinds of bodies that gravitated in different proportions to their quantities of matter, H3 would entail they could transform into one another and "take on all the intermediate degrees" of gravity. If this were the case, however, we would expect to find variation in the gravitational response of bodies of different kinds (wood, gold, wheat, etc.). Since we do not, there must not 21As Mary Domski pointed out in conversation, this claim already involves an inductive leap from all bodies on which experiments have been made to all bodies on which experiments can be made. Even for a single body, the claim that weight is proportional to quantity of matter requires a leap from measurements that have been made to all possible measurements. In general, measurement and induction are intertwined. As Newton wrote, on induction "all experimental philosophy is founded" (Koyré 1965, 269). 22For H3's alchemical and neo-platonic background, see Dobbs (1991, Ch. 3) and McGuire (1970). Rule 3: Induction and Universality 11 be kinds of bodies that gravitate in different proportions to their quantities of matter - "all bodies universally" must gravitate in the same way (Principia, 809).23 Several issues surround R3 and H3, of which I can address only a few.24 To begin with, we might ask what Newton meant by "all bodies universally" or "universal gravity." The question is important, as Newton also asserted that gravity was not essential to bodies. Some have claimed that Newton thought gravity was a necessary property of bodies, or perhaps a primary property. Others have claimed that Newton really did think gravity was essential, but wished to avoid controversy.25 A common implication is that Newton chose "universal" as a placeholder for a more loaded term, or perhaps was just imprecise about the variety of ways in which properties can be associated with their bearers. But this is wrong. Questions concerning gravity's metaphysical status are apropos, of course - and they were immediately raised by Newton's contemporaries - but Newton's use of 'universality' mirrored common technical uses and was quite natural. It was also deflationary, in the sense that it did not directly entail any position regarding gravity's essentiality, necessity, or primacy. In the late seventeenth-century, it was not unusual to think of universal qualities (or propositions, principles, causes, etc.) as those that applied to every member of a given class, without necessarily implying more loaded ontic/epistemic terms. "Universal" (and cognates) was used synonymously with "general" and "common," as Newton used it in R3: The qualities of bodies that cannot be intended and remitted [etc.]. . . should be taken as qualities of all bodies universally [qualitatibus corporum universorum]. For the qualities of bodies. . . that square with experiments universally [generaliter ] are to be regarded as universal [generales] qualities. . . (Principia, 795) Knowledge of such general qualities (propositions, principles, causes, etc.) was frequently taken to be the result of induction from particulars.26 For example, Samuel Smith wrote in a book Newton heavily annotated that induction proceeds "A singularibus ad universale."27 Robert Sanderson, in another book Newton knew well, held 23H3's gradualism is essential. Denying it but holding on to transmutation can entail, for example, that non-gravitating bodies may transform without any intermediate steps into bodies that gravitate in direct proportion to their quantities of matter. In such a case, there could be no change to the observable proportion of gravity to quantity of matter, and so evidence about gravitating bodies would not entail anything interesting about non-gravitating ones. Gradualism eliminates this possibility by demanding detectable variation. 24I will not address the intellectual background to the intention/remission criterion, the connection of R3 to atomism and mechanism, H3's relation to metaphysics of 'the great chain of being,' the differences between universality and essentiality, the idea of treating gravity as a quality vs. an interaction, Newton's appeal to the experimentum crucis, and the logical status of transduction (i.e., induction to in-principle unobservable entities). The most thorough treatments of H3/R3 are still McGuire (1967), McGuire (1968b), McGuire (1968a), and McGuire (1970). 25For different approaches, see Koyré (1965, 149ff), McGuire (1968b), McMullin (1978), Janiak (2008), Schliesser (2009), Henry (1994), Ducheyne (2014). 26The connection between induction and particulars traces to Aristotle, e.g., Topics I.12. Similar ideas were repeated in the seventeenth-century, without any necessary commitment to induction being the only way to arrive at universals. See Arnauld and Nicole (1696); Spinoza ([1677] 1985, IIp40s1); Hobbes (1656, Lesson 5). Some disputed that induction can result in universals, e.g., Leibniz (1969, 129–30). See Milton (1987) and Bolton (2003). 27S. Smith (1656, 111ff, 144ff, 154ff). Sgarbi (2013, 156) notes: "Sanderson emphasizes particularly the extreme utility of induction for discovering first principles and universals of the causes and of 12 Newton's Regulae Philosophandi that induction is the process "by which we make up a universal conclusion summoning many experiences" (quoted in Mamiani 2001, 12). Importantly, induction from particulars, then as now, was an inference to some relevant class, not necessarily to all objects in the universe. Newton too used "universal" to capture members of restricted classes. For example: "near the surface of the earth. . . the accelerative gravity . . . is the same in all bodies universally [in corporibus universis]," and "there is gravity toward all planets universally [in planeta universa]" (Principia, 408). The first class concerns only objects near the surface of the earth, the second only primary and secondary planets (fewer than 20 objects). English translations of universus (and cognates) are particularly misleading, as they tend to render the single adjective as a determiner and an adverb: "all . . . universally." This might suggest that "universally" adds a layer of meaning to "all," but it doesn't. The adverb is sometimes dropped altogether; e.g., "Therefore the sun gravitates toward all the primary and secondary planets [planetas universos], and all these [universi ] toward the sun" (Principia, 390). In general, universality connoted application to every member of a class, but no more. It did not entail, at least not generally, a position about whether the universality of x rested on x 's essentiality, necessity, etc. Thus, when Newton asserted that gravity is universal, he was just asserting that it is attributable to every body within the relevant class of bodies. His incredulity at accusations that he had done otherwise is understandable. Moreover, given the well-discussed nature of induction, Newton would not have expected - nor did he receive - objections regarding the validity of inductive inference itself. Rather, the objections he expected - and received - concerned how to delimit the relevant class of bodies in III.6, i.e, the induction's scope. We can see now why H3 provided the initial justification for gravity's universality, and why the argument from transmutation remained in III.6.c2 after H3 was removed. The transmutation hypothesis asserted that despite the apparent multiplicity of kinds of bodies in common experience and despite the common belief among vortex theorists that the insensibly small constituents of matter lacked gravity, there were no insuperable differences between types of bodies. As Newton put it in a draft conclusion to E1: "the matter of all things is one and the same, which is transmuted into countless forms by the operations of nature" (Newton 1962, 341). H3 provided an explicit reason for taking the relevant inductive class in III.6 to be the class of all bodies.28 Before addressing the homogeneity of matter further, we might ask why Newton abandoned H3 if he retained the argument it supported. Opinions vary. I. B. Cohen suggests that H3 was too vulnerable to criticism by supporters of alternate matter theories (Cohen 1999, 203). Ernan Mcmullin suggests that Newton came to realize that H3 conflicted with atomism, since it allowed even mechanical qualities like impenetrability to be transmuted (McMullin 1978, 143.22). J. E. McGuire defends the compatibility of H3 with atomism - a position I assumed above - but suspects that Newton didn't want to defend his atomism publicly. McGuire also argues that in the early 1690s (when Newton met repeatedly with John Locke and Lady Masham) Newall other universal things to be proved. But he also recognizes the intrinsic weakness of induction, in that a single exception or counterexample can overturn its conclusions." 28Newton also had more technical reasons for asserting the size-invariability of the universal qualities of matter; see Belkind (mss) and Schliesser and G. E. Smith (forthcoming). Rule 3: Induction and Universality 13 ton adopted Locke's primary/secondary distinction and realized the distinction was incompatible with a literal reading of H3 (McGuire 1967). Alan Shapiro suggests that H3 could not properly certify the qualities to which Newton appealed in optical explanations (Shapiro 1993, 40ff). For the most part, these are compatible positions, at least insofar as they emphasize Newton's move from the explicit ontological commitments of H3 to the more methodological approach of R3. I suspect, however, that there is also another, more specifically aimed reason for Newton's formulation of R3. Once again, the rule's context of composition is informative. Still called an Hypothesis, R3 was first mentioned in an errata sheet to E1, dated 13 March 1689/90 by Fatio de Duiller, a young mathematician who was Newton's confidant at the time. Fatio was also a a close associate of Christiaan Huygens, a Dutch mathematician of international standing. In the previous summer, Newton had met with Huygens to discuss, among other things, the Dutchman's soon-to-be-published Discourse on the Cause of Gravity, which contained an "addition" challenging the Principia. In February of 1689/90, Fatio indicated to Newton that he was soon to receive a copy of the work, and suggested that "it being writ in French you may perhaps choose rather to read it here [London] with me" (Correspondence, V 390). Newton left Cambridge on March 10th, and likely spent the month in London with Fatio, reading Huygens's work. The errata sheet dated 13 March was later sent to Huygens and ended up in Leibniz's Hanover manuscripts (Cohen 1971, §VII.10). Huygens's Discourse began with: In order to find an intelligible cause of gravity we must see how gravity can come about while presupposing in nature only bodies that are made from a like matter, and considering in these neither a quality nor a tendency (aucune qualité ni aucune inclination) to draw near one another, but only their different magnitudes, figures, and motions. . . We judge rightly from the start that there is nothing in the appearance attributed to the figure, or in the smallness of its corpuscles, with an effect that resembles that of gravity, which, being an endeavor or a tendency to motion, must in all likelihood be produced by a motion. So, there remains only to find in what manner it can act, and in which bodies it can be encountered . . . (Huygens 1690a, 451, emphasis added). Huygens, a Cartesian, did not find any problem with Newton's position on the uniformity of matter. He believed, however, that "modern authors" triumphed when they rejected "some internal and inherent quality that makes [bodies] tend downward. . . or a tendency for the parts to join together" (Huygens 1690a, 455). Clear thinking showed that matter was purely mechanical and so only mechanical qualities could be used in genuine physical - i.e., causal - explanations. Huygens toed the Cartesian line in asserting that weight was not such a quality; it was not possessed by all bodies and so could not be appealed to as a genuine cause. But, importantly, Huygens also opposed Descartes. He included in his list of mechanical attributes "perfect hardness" that renders fundamental bodies "[1] impenetrable and [2] incapable of being. . . broken" (Huygens 1690a, 473). This claim was made in a paragraph about Newton's rejection of vortices - no doubt Newton took note of it. Although this is no smoking gun, R3 seems like a direct rebuttal to Huygens. To begin with, the rule does not offer an explicit defense of the uniformity of matter (as H3 did), since this was a point of agreement. Instead, it focuses on which features of bodies can be "declared general" and thus be used in physical explanations. Even in 14 Newton's Regulae Philosophandi the initial version Fatio transcribed, the rule already stressed experience and induction: "the laws and properties on which it is possible to institute experiments are the laws and properties of all bodies whatsoever (universorum)" (Huygens 1950, X 155; McGuire 1968b, 241). In all versions, Newton argued that the rule explained why mechanical qualities were valued to begin with, and so revealed the true, non-speculative grounds of Huygens's position. That Huygens was a target is most clearly implied by Newton's caveats. First, he made a point of arguing that [2] was an overreach: the indivisibility endorsed by Huygens did not satisfy the criterion, since there was no phenomena corresponding to it. He even used the occasion to lecture on the value of the experimentum crucis for disproving indivisibility, perhaps a methodological grievance carried over from his optical debates with Huygens.29 Second, he hoisted Huygens by his own petard by noting that "the argument from phenomena will be even stronger for universal gravity than for [1] the impenetrability of bodies," precisely that feature of bodies that Huygens proudly championed against Descartes! Whether or not R3 was aimed at Huygens, comparing it to the Discourse reveals why gravity's universality was important to establish. By showing that weight (more precisely, weight-towards) was attributable to every body and on par with extension and motion, Newton licensed its reintroduction - contra Huygens - into genuine physical explanations. We should take seriously Newton's claim that R3 provided "the foundation of all natural philosophy" (Principia, 796, emphasis added). The rule furnished the physical vocabulary for fundamental natural philosophical explanations. That is, it provided a roster of those basic features of the world that could ground the synthetic portion of Newton's regressus, a roster of those features that could be taken as genuine properties and causes. Of course, as Galileo's introduction of time into the study of free-fall opened a range of questions and answers that was unavailable to his predecessors, so Newton's reintroduction of weight opened a range of questions and answers that was unavailable to mechanical philosophers. For the many (like Régis) who could not understand the complexity of Newton's mathematics or his empirical arguments, this was the Principia's main claim. To close our discussion of R3, let's return briefly to the homogeneity of matter. It is important to appreciate the extent to which it supported several of the Principia's inferences, not only the inductive generalization of III.6. For example, in proposition III.5 - the proposition that elicited R4 - homogeneity supported the extension of gravity from planets with moons to planets without moons: "[f]or no one doubts that Venus, Mercury, and the rest are bodies of the same kind as Jupiter and Saturn" (Principia, 806).30 In the Classical Scholia to propositions III.4–9, the homogeneity of matter was a repeated theme: "For all bodies, inspite of appearances, are composed of identical parts and of a single material" (Newton 2001, 229). Most importantly, in The System of the World - Newton's preliminary version of Book III - there were no analogs to the Hypotheses or Regulae, yet the "universal nature of matter" played an explicit justificatory role (Newton 1727, §25). On what ground did Newton assert the homogeneity of matter? In R3 and its drafts, Newton cited "the analogy of nature" - the idea that "nature is always simple and ever 29The 1690 Discourse was printed alongside Huygen's Traité de la Lumière. 30In an annotated copy of E1, Newton explicitly backed III.5.c2 "per Hypoth III." Rules 1 & 2: Simplicity and Uniformity 15 consonant with itself" (Principia, 795). Once again, a full discussion is impossible.31 However, the simplicity of nature turns our attention to R1 and R2, as it was also their explicit justification. 1.4 Rules 1 & 2: Simplicity and Uniformity Rule 1 No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena. . . . For nature is simple and does not indulge in the luxury of superfluous causes. Rule 2 Therefore, the causes assigned to natural effects of the same kind must be, so far as possible the same. R1 and R2 are principles of causal simplicity and uniformity that remained virtually identical in all editions of the Principia.32 They are also the most cited. In all editions, R2 is cited in support of III.5, and in E3, both R1 and R2 are cited in support of III.4 and III.5. Proposition III.4 shows that the force holding the moon in its orbit is centripetally directed at the earth and has the same strength as the force associated with free-fall, to a high degree of precision. Given this similarity of mathematical attributes, and following R1's demand for parsimony of causes, the proposition concludes that the two forces must actually be one.33 III.5 is likewise premised on the similarity of mathematical attributes of the forces holding the moons of Jupiter and Saturn in their orbits and the force holding the earth's moon in its orbit: all are centripetal and inverse square. Following R2's demand for parsimony of kinds of causes, the proposition concludes that the forces must be of the same kind, all instances of gravity. Both inferences go to eliminate the possibility of forces that are mathematically similar but belong to different physical kinds. In III.4's case, we might imagine that objects on earth are drawn by a force that affects terrestrial objects, but not the moon; and the moon is drawn by a force that affects moon-matter, but not terrestrial objects. In III.5's case, we might imagine a Jupiter-force and a Saturn-force that affect Jupitermoon-matter and Saturn-moon-matter, but cannot affect anything else. R1 and R2 advise against taking such possibilities seriously, since nature is "simple and does not indulge in the luxury of superfluous causes." It is more reasonable to suppose a paucity than a diversity of forces. Moreover, it is more reasonable to suppose a paucity of kinds of forces. 31See, e.g., McGuire (1970), Okruhlik (1989). 32Their most significant revision was the introduction of "so far as possible" to R2 in E3. Ducheyne (2012, Ch. 5) puts the addition in the context of Newton's overall 'epistemic' turn in E3. 33Matters are more complicated in E1. In that edition, no rules are cited in support of III.4, and only R2 (then H2) is cited in support of III.5. In fact, R1 (then H1), is not cited at all. Although I cannot defend the following conjecture here, I believe that H1's curious status indicates that Newton did not initially see it as the relevant principle for III.4; or, at least, saw it as primarily applying elsewhere. Specifically, I believe that H1 was intended to support the mutuality of gravitation. It recommends taking the attraction between two bodies not as two forces - one body acting of the other, and the other acting on the one - but as one action that "is simple and single [simplex est et unica]." That inference was highlighted in Newton (1727, §21), but was more submerged by E1. 16 Newton's Regulae Philosophandi Importantly, identifying forces also entails identifying types of matter. If the force affecting bodies on earth is capable of affecting the moon, earth-matter and moonmatter must be relevantly similar.34 The converse is also true. If the earth and moon are made of the same matter, a single force should be able to affect both (ceteris paribus). Similarly, if Jupiter and Saturn are made of the same matter, there isn't an obvious sense in which to maintain a distinction between a Jupiter-force and a Saturnforce. The principle of causal parsimony is thus bound up with the homogeneity of matter.35 In the same way, it is bound up with the idea that natural operations are uniform in space, time, and across non-fundamental genera. As Newton wrote in the explanatory text to R2: Examples [of causes that are likely to be of the same kind] are the cause of respiration in man and beast, or of the falling of stones in Europe and America, or of the light of a kitchen fire and the sun, or the reflection of light on our earth and the planets. (Principia, 795) But why should we suspect that the cause of the falling is the same in Europe and America? Because we already believe that Europe and America are relevantly alike. And, sotto voce, because Europe and America are relevantly alike, we are justified in thinking that the cause of falling in them is the same. In other words, uniformity is what allows parsimony to be an effective methodological strategy. Moreover, uniformity is what allows induction to be an effective methodological strategy. It should be equally apparent - if only from Newton's direct appeal to Nature's consonance in R3 - that the "Rule of Induction" also relies on spatial and temporal uniformity. Without it, Newton's generalization from bodies "on which experiments can be made" to distant bodies, bodies at any scale, and future bodies would be rash.36 Of course, uniformity cannot guarantee the success of any given induction; hence R4. R1 and R2 thus express a commitment to natural simplicity cum uniformity cum homogeneity which R3 and R4 presume. Their central role did not go unnoticed. None other than David Hume - one rather sensitive to the presuppositions of induction - wrote that R2 was "Newton's chief rule of philosophizing" (Hume [1751] 1998, §3.48).37 Perhaps he was following Roger Cotes, who in the preface to E2 discreetly disputed Newton's characterization of R3 as the "foundation of all philosophy" by claiming that "All philosophy is based on this rule [R2], inasmuch as, if it taken away, there is then nothing we can affirm about things universally" (Principia, 391, emphasis added). At any rate, Newton's deep commitment to natural simplicity, uniformity, and heterogeneity is a central and repeated theme in his work. He asserted it throughout his life, 34The classical scholium to III.4 begins with "That the earth's moon is a dense body made of earth." Newton (2001, 219). 35III.4 offers an additional 'empirical' argument, which some believe stands independently of causal parsimony, and thus renders R1 and R2 superfluous: If the forces in question were not identical, two forces would affect bodies, and thus bodies would fall twice as fast; contrary to experience. I believe this fails to appreciate the extent to which the homogeneity of matter is involved in the 'empirical' argument, and thus the extent to which a substantive assumption about natural uniformity is nevertheless required; see Di Fate (2011). 36In the Opticks, Newton speculates that different types of matter and different laws may exist in different parts of the universe. For the compatibility of this claims with R3, see Biener and Schliesser (2017). 37De Pierris (2006) has argued that this is a reference to R3, but Hazony and Schliesser (2015) argue for the present interpretation. Conclusion 17 often playing on the same phrase: "Nature is ever simple and conformable to herself" (e.g., unpublished preface to the Principia (Newton 1962, 307), draft conclusion to the Principia (Newton 1962, 333), "Di vi electrica" (Correspondence, V 366), and the Opticks passages quoted above.) Here, finally, our discussion must reach past the rules. Although Newton once suggested that the rules are themselves the products of induction, his commitment to Nature's simplicity is more likely due to his ardent theological convictions.38 Newton believed that the world was created by a providential God in a way that allowed His servants to understand it, and, in consequence, to understand something of Him. Of course, created beings cannot understand everything about the world. Newton was clear that knowledge of essences, for example, was impossible (Principia, 942; Newton 2004a, 27). However, as the General Scholium and letters to Bentley demonstrate he was deeply committed to arguments from design, and these presupposed that God has made the world so that His design is at least partially apparent (Principia, 940–3; Newton 2004b, 94–7).39 The link between God, His design, and simplicity was made plain by Newton some ten years before composing the Principia. In a treatise on revelation, he wrote: As the world, which to the naked eye exhibits the greatest variety of objects, appears very simple in its internall constitution when surveyed by a philosophic understanding, and so much the simpler by how much the better it is understood. . . It is the perfection of God's works that they are all done with the greatest simplicity. He is the God of order and not confusion. And therefore as they that would understand the frame of the world must indeavour to reduce their knowledg to all possible simplicity, so it must be in seeking to understand these visions. (Untitled Treatise on Revelation (§1.1), 14r, Newton Project, quoted in Snobelen 2005, 234). The perfection of God is demonstrated in His works, and their perfection is demonstrated in their simplicity and order. All in all, it is God's relation to creation that makes the Rules of Philosophizing reasonable precepts to follow. That relation is the tacit premise behind them and, indeed, behind Newton's physical investigations. Without it, there would be no sense to the Rules and no sense to natural philosophy. 1.5 Conclusion While addressing various of smaller issues, I have traced a line of thought through Newton's four rules to the substantive assumptions about God (and thus Nature) that underlay them. Although we can spend much more time on the character of Newton's theological commitments, the point should be clear. Newton believed that Nature is simple and orderly because it is divinely decreed, and that, consequently, human beings can reason about it effectively. In particular, they can (fallibly) determine causes and make inductive generalizations on which they can base natural philosophical knowledge. I have also tried to describe the highly interrelated nature of the rules and the ways in which Newton often used different rules for one and the same purpose, as 38In a revision to E2's rules, Newton wrote "A Rule I call every Proposition that is (established from) gathered from Phenomena through the argument of Induction and agrees with them" (quoted in McGuire 1970, 70). This seems to have been an errant thought, quickly abandoned. 39I thank John Henry for stressing this to me. 18 Newton's Regulae Philosophandi well as one and the same rationale for different rules. I believe this interrelatedness shows that the rules were less distinct in Newton's mind than they appear, and that, consequently, they are differing expressions of the fundamental idea outlined above. To close, I'd like to point out that despite Newton's thorough reorientation of the methods and presuppositions of physico-mathematics, his broader methodological outlook vis-à-vis the rules, particularly their relation to theology, was not unusual. The following passage is from Isaac Barrow, Newton's mentor and first Lucasian professor. It anticipates virtually all of the rules and demonstrates their tight interrelations. Moreover, it clearly ties questions of methodology to premises about God. It even makes a failure to abide by these methodological precepts kin to mortal sin! No doubt Newton approved. [W]here any Proposition is found agreeable to constant Experience, especially where it seems not to be conversant about the Accidents of Things, but pertains to their principal Properties and intimate Constitution, it will at least be most safe and prudent to yield a ready Assent to it. For as we are justly accused of a rash Temerity, by suffering ourselves to be so much as once deceived by our Faith, so we are guilty of the greatest Imprudence, if we shew the least Distrust, and do not yield our stedfast Assent and obstinately adhere, when we still find our Expectations answered as accurately as possible (quam accuratissime), after a thousand Researches; and especially when we have the constant Agreement of Nature to confirm our Assent, and the immutable Wisdom of the first Cause forming all Things according to simple Ideas, and directing them to certain Ends: Which Consideration alone is almost sufficient to make us look upon any Proposition confirmed with frequent Experiments, as universally true (universaliter vera), and not suspect that Nature is inconstant and the great Author of the Universe unlike himself. 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