Natural Histories and Newton’s Theory of the Tides
Kirsten Walsh writes…
Lately, I’ve been thinking about Newton’s work on the tides. In the Principia Book 3, Newton identified the physical cause of the tides as a combination of forces: the Moon and Sun exert gravitational pulls on the waters of the ocean which, together, cause the sea levels to rise and fall in regular patterns. This theory of the tides has been described as one of the major achievements of Newtonian natural philosophy. Most commentators have focussed on the fact that Newton extended his theory of universal gravitation to offer a physical cause for the tides—effectively reducing the problem of tides to a mathematical problem, the solution of which, in turn, provided ways to establish various physical features of the Moon, and set the study of tides on a new path. But in this post, I want to focus on the considerable amount of empirical evidence concerning tidal phenomena that underwrites this work.
Let’s begin with the fact that, while Newton’s empirical evidence of tidal patterns came from areas such as the eastern section of the Atlantic Ocean, the South Atlantic Sea, and the Chilean and Peruvian shores of the Pacific Ocean, Newton never left England. So where did these observational records come from?
Newton’s data was the result of a collective effort on a massive scale, largely coordinated by the Royal Society. For example, one of the earliest issues of the Philosophical Transactions published ‘Directions for sea-men bound for far voyages, drawn up by Master Rook, late geometry professour of Gresham Colledge’ (1665: 140-143). Mariners were instructed “to keep an exact Diary [of their observations], delivering at their return a fair Copy thereof to the Lord High Admiral of England, his Royal Highness the Duke of York, and another to Trinity-house to be perused by the R. Society”. With respect to the tides, they were asked:
“To remark carefully the Ebbings and Flowings of the Sea, in as many places as they can, together with all the Accidents, Ordinary and Extraordinary, of the Tides; as, their precise time of Ebbing and Flowing in Rivers, at Promontories or Capes; which way their Current runs, what Perpendicular distance there is between the highest Tide and lowest Ebb, during the Spring-Tides and Neap-Tides; what day of the Moons age, and what times of the year, the highest and lowest Tides fall out: And all other considerable Accidents, they can observe in the Tides, cheifly neer Ports, and about Ilands, as in St. Helena’s Iland, and the three Rivers there, at the Bermodas &c.”
This is just one of many such articles published in the early Philosophical Transactions that articulated lists of queries concerning sea travel, on which mariners, sailors and merchants were asked to report. In its first 20 years, the journal published scores of lists of queries relating to the tides, and many more reports responding to such queries. This was Baconian experimental philosophy at its best. The Royal Society used its influence and wide-ranging networks to construct a Baconian natural history of tides: using the method of queries, they gathered observational data on tides from all corners of the globe which was then collated and ordered into tables.
Newton’s engagement with these observational records is revelatory of his attitudes and practices relating to Baconian experimental philosophy. Firstly, especially in his later years, Newton was regarded as openly hostile towards natural histories. However, here we see Newton explicitly and approvingly engaging with natural histories. For example, in his discussion of proposition 24, he drew on observations by Samuel Colepresse and Samuel Sturmy, published in the Philosophical Transactions in 1668, explicitly offered in response to queries put forward to John Wallis and Robert Boyle in 1665:
“Thus it has been found by experience that in winter, morning tides exceed evening tides and that in summer, evening tides exceed morning tides, at Plymouth by a height of about one foot, and at Bristol by a height of fifteen inches, according to the observations of Colepress and Sturmy” (Newton, 1999: 838).
Colepresse’s Tidal Scheme for Plymouth, 1667
I have argued previously that Newton was more receptive to natural histories than is usually thought. The case of the tides offers additional support for my argument. Newton’s notes and correspondence show that, from as early as 1665, he was heavily engaged in the project of generating a natural history of the tides, although he never contributed data. And eventually, he was able to use these empirical records to theorise about the cause of the tides. This suggests that Newton didn’t object to using natural histories as the basis for theorising. Rather, he objected to treating natural histories as the end goal of the investigation.
Secondly, I have previously discussed the fact that Newton seldomly reported ‘raw data’. The evidence he provided for Phenomenon 1, for example, included calculated average distances, checked against the distances predicted by the theory. Newton’s empirical evidence on the tides, as reported in the Principia, was similarly manipulated and adjusted with reference to his theory. Commentators have largely either condemned or ignored this ‘fudge factor’, but such adjustments are ubiquitous in Newton’s work, suggesting that they were a key aspect of his practice. Newton recognised that ‘raw data’ had limited use: to be useful, data needed to be analysed and interpreted. In short, it needed to be turned into evidence. The Baconians appear to have recognised this: queries guide the collection of data, which is then ordered into tables in order to reveal patterns in the data. As this case makes clear, however, Newton’s theory-mediated manipulation of the data went beyond basic ordering, drawing on causal assumptions to reveal phenomena from the data.
Thirdly, this case emphasises Newton’s science as embedded in rich social, cultural and economic networks. The construction of this natural history of tides was an organised group effort. That Newton had access to data collected from all over the world was the result of hard work from natural philosophers, merchants, mariners and priests who participated in the accumulation, ordering and dissemination of this data. Further, the capacities of that data to be collected itself followed the increasingly global trade networks reaching to and from Europe. Newton’s work on the tides was the very opposite of a solitary effort.
On this blog, we have noted in passing, but not explored in depth, the crucial roles played by travellers’ reports and information networks in Baconian experimental philosophy. Newton’s study of the tides is revelatory of the attitudes and practices of early modern experimental philosophers with respect to such networks. I shall discuss these in my next post.
Understanding Newton’s Principia as part of the Baconian Tradition
Kirsten Walsh writes…
Lately I have been examining Baconian interpretations of Newton’s Principia. First, I demonstrated that Newton’s Moon test resembles a Baconian crucial instance. And then, I demonstrated that Newton’s argument for universal gravitation resembles Bacon’s method of gradual induction. This drew our attention to some interesting features of Newton’s approach, bringing the Principia’s experimental aspects into sharper focus. But they also highlighted a worry: Newton’s methodology resembling Bacon’s isn’t enough to establish that Newton was influenced by Bacon. Bacon and Newton were gifted methodologists—they could have arrived independently at the same approach. One way to distinguish between convergence and influence is to see if there’s anything uniquely or distinctively Baconian in Newton’s use of crucial experiments and gradual induction. Another way would be if we could find some explicit references to Bacon in relation to these methodological tools. Alas, so far, my search in these areas has produced nothing.
In this post, I’ll consider an alternative way of understanding Baconianism in the Principia. I began this series by asking whether we should regard Newton’s methodology as an extension of the Baconian experimental method, or as something more unique. In answering, I have hunted for evidence that the Principia is Baconian insofar as Newton applied Baconian methodological tools in the Principia. But you might think that whether Newton was influenced by Bacon isn’t so relevant. Rather, what matters is how the Principia was received by Newton’s contemporaries. So in this post, I’ll examine Mary Domski’s argument that the Principia is part of the Baconian tradition because it was recognised, and responded to, as such by members of the Royal Society.
Domski begins by dispelling the idea that there was no place for mathematics in the Baconian experimental tradition. Historically, Bacon’s natural philosophical program, centred on observation, experiment and natural history, was taken as fundamentally incompatible with a mathematical approach to natural philosophy. And Bacon is often taken to be deeply distrustful of mathematics. Domski argues, however, that Bacon’s views on mathematics are both subtler and more positive. Indeed, although Bacon had misgivings about how mathematics could guide experimental practice, he gave it an important role in natural philosophy. In particular, mathematics can advance our knowledge of nature by revealing causal processes. However, he cautioned, it must be used appropriately. To avoid distorting the evidence gained via observation and experiment, one must first establish a solid foundation via natural history, and only then employ mathematical tools. In short, Bacon insisted that the mathematical treatment of nature must be grounded on, and informed by, the findings of natural history.
Domski’s second move is to argue that seventeenth-century Baconians such as Boyle, Sprat and Locke understood and accepted this mathematical aspect of Bacon’s methodology. Bacon’s influence in the seventeenth century was not limited to his method of natural history, and Baconian experimental philosophers didn’t dismiss speculative approaches outright. Rather, they emphasised that there was a proper order of investigation: metaphysical and mathematical speculation must be informed by observation and experiment. In other words, there is a place for speculative philosophy after the experimental stage has been completed.
Domski then examines the reception of Newton’s Principia by members of the Royal Society—focusing on Locke. For Locke, natural history was a necessary component of natural philosophy. And yet, Locke embraced the Principia as a successful application of mathematics to natural philosophy. Domski suggests that we read Locke’s Newton as a ‘speculative naturalist’ who employed mathematics in his search for natural causes. She writes:
[O]n Locke’s reading, Newton used a principle—the fundamental truth of universal gravitation—that was initially ‘drawn from matter’ and then, with evidence firmly in hand, he extended this principle to a wide store of phenomena. By staying mindful of the proper experimental and evidentiary roots of natural philosophy, Newton thus succeeded in producing the very sort of profit that Sprat and Boyle anticipated a proper ‘speculative’ method could generate (p. 165).
In short, Locke regarded Newton’s mathematical inference as the speculative step in the Baconian program. That is, building on a solid foundation of observation and experiment, Newton was employing mathematics to reveal forces and causes.
In summary, Domski makes a good case for viewing the mathematico-experimental method employed in the Principia as part of the seventeenth-century Baconian tradition. I have a few reservations with her argument. For one thing, ‘speculative naturalist’ is surely a term that neither Locke nor Newton would have been comfortable with. And for another thing, although Domski has provided reasons to view Newton’s mathematico-experimental method as related to, and a development of, the experimental philosophy of the Royal Society, I’m not convinced that this shows that they viewed the Principia as Baconian. That is to say, there’s a difference between being part of the experimental tradition founded by Bacon, and being Baconian. I’ll discuss these issues in my next post, and for now, I’ll conclude by discussing some important lessons that I think arise from Domski’s position.
Firstly, we can identify divergences between Newton and the Baconian experimental philosophers. And these could be surprising. It’s not, in itself, his use mathematics and generalisations that makes Newton different—Domski has shown that even the hard-out Baconians could get on board with these features of the Principia. The differences are subtler. For example, as I’ve discussed in a previous post, Boyle, Sprat and Locke advocated a two-stage approach to natural philosophy, in which construction of natural histories precedes theory construction. But Newton appeared to reject this two-stage approach. Indeed, in the Principia, we find that Newton commences theory-building before his knowledge of the facts was complete.
Secondly, the account highlights the fact that early modern experimental philosophy was a work in progress. There was much variation in its practice, and room for improvement and evolution. Moreover, its modification and development was, to a large extent, the result of technological innovation and the scientific success of works like the Principia. Indeed, it was arguably the ability to recognise and incorporate such achievements that allowed experimental philosophy to become increasingly dominant, sophisticated and successful in the eighteenth century.
Thirdly, the account suggests that, already in the late-seventeenth century, the ESD framework was being employed to guide, and also to distort, the interpretation and uptake of natural philosophy. By embracing the Principia as their own, the early modern experimental philosophers intervened on and shaped its reception, and hence, the kind of influence the Principia had. This raises an interesting point about influence.
As I have already noted, it is difficult to establish a direct line of influence stretching from Bacon to Newton. But, by focusing on how Bacon’s program for natural philosophy was developed by figures such as Boyle, Sprat and Locke, we can identify a connection between Bacon’s natural philosophical program and Newton’s mathematico-experimental methodology. That is, we can distinguish between influence in terms of actual causal connections—Newton having read Bacon, for instance—and influence insofar as some aspect of Newton’s work is taken to be related to Bacon’s by contemporary (or near-contemporary) thinkers. Indeed, Newton could have been utterly ignorant of Bacon’s actual views on method, but the Principia might nonetheless deserve to be placed alongside Bacon’s work in the development of experimental philosophy. Sometimes what others take you to have done is more important than what you have actually done!
Baconian Induction in the Principia
Kirsten Walsh writes…
Recently, I have been looking for clear cases of Baconianism in the Principia. In my last post, I offered Newton’s ‘moon test’ as an example of a Baconian crucial instance, ending with a concern about establishing influence between Bacon and Newton. Newton used his calculations of the accelerations of falling bodies to provide a crucial instance which allowed him to choose between two competing explanations. However, one might argue that this was simply a good approach to empirical support, and not uniquely Baconian. In this post, I’ll consider another possible Baconianism: Steffen Ducheyne’s argument that Newton’s argument for universal gravitation resembles Baconian induction.
Let’s begin with Baconian induction (this account is based on Ducheyne’s 2005 paper). Briefly, Bacon’s method of ampliative inference involved two broad stages. The first was a process of piecemeal generalisation. That is, in contrast to simple enumerative induction, shifting from the particular to the general in a single step, Bacon recommended moving from particulars to general conclusions via partial or mediate generalisations. Ducheyne refers to this process as ‘inductive gradualism’. The second stage was a process of testing and adjustment. That is, having reached a general conclusion, Bacon recommended deducing and testing its consequences, adjusting it accordingly.
Ducheyne argues that, in the Principia, Newton’s argument for universal gravitation proceeded according to Baconian induction. In the first stage, Newton’s argument proceeded step-by-step from the motion of the moon with respect to the Earth, the motions of the moons of Jupiter and Saturn with respect to Jupiter and Saturn, and the motions of the planets with respect to the Sun, to the forces producing those motions. He inferred that the planets and moons maintain their motions by an inverse square centripetal force, and concluded that this force is gravity—i.e. the force that causes an apple to fall to the ground. And, in a series of further steps (still part of the first stage), Newton established that, as the Sun exerts a gravitational pull on each of the planets, so the planets exert a gravitational pull on the Sun. Similarly, the moons exert a gravitational pull on their planets. And finally, the planets and moons exert a gravitational pull on each other. He concluded that every body attracts every other body with a force that is proportional to its mass and diminishes with the square of the distance between them: universal gravitation. Moving to the second stage, Newton took his most general conclusion—that gravity is universal—and examined its consequences. He demonstrated that the irregular motion of the Moon, the tides and the motion of comets can be deduced from his theory of universal gravitation.
Ducheyne notes that Newton didn’t attribute this method of inference to Bacon. Instead, he labelled the two stages ‘analysis’ and ‘synthesis’ respectively, and attributed them to the Ancients. However, Ducheyne argues that we should recognise this approach as Baconian in spirit and inspiration.
This strikes me as a plausible account, and it illuminates some interesting features of Newton’s approach. For one thing, it helps us to make sense of ‘Rule 4’:
In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions.
Newton’s claim that, in the absence of counter-instances, we should take propositions inferred via induction to be true seems naïve when interpreted in terms of simple enumerative induction. However, given Newton’s ‘inductive gradualism’, Rule 4 looks less epistemically reckless.
Moreover, commentators have often been tempted to interpret this rule as an expression of the hypothetico-deductive method, in which the epistemic status of Newton’s theory is sensitive to new evidence. Previously, I have argued that, when we consider how this rule is employed, we find that it’s not the epistemic status of the theory, but its scope, that should be updated. Ducheyne’s Baconian interpretation supports this position—and perhaps offers some precedent for it.
Ducheyne’s suggestion also encourages us to re-interpret other aspects of Newton’s argument for universal gravitation in a Baconian light. Consider, for example, the ‘phenomena’. Previously, I have noted that these are not simple observations but observed regularities, generalised by reference to theory. They provide the explananda for Newton’s theory. In Baconian terms, we might regard the phenomena as the results of a process of experientia literata—they represent the ‘experimental facts’ to be explained. This, I think, ought to be grist for Ducheyne’s mill.
Interpreting Newton’s argument for universal gravity in terms of Baconian induction brings the experimental aspects of the Principia into sharper focus. These aspects have often been overlooked for two broad reasons. The first is that the mathematical aspects of the Principia have distracted people from the empirical focus of book 3. I plan to examine this point in more detail in my next post. The second is that the Baconian method of natural history has largely been reduced to a caricature, which has made it difficult to recognise it when it’s being used. Dana Jalobeanu and others have challenged the idea that a completed Baconian natural history is basically a large storehouse of facts. Bacon’s Latin natural histories are complex reports containing, not only observations, but also descriptions of experiments, advice and observations on the method of experimentation, provisional explanations, questions, and epistemological discussions. We don’t find such detailed observation reports in the Principia, but we do find some of the features of Baconian natural histories.
So, Ducheyne’s interpretation of Newton’s argument for universal gravitation in terms of Bacon’s gradualist inductive method proves both fruitful and insightful. However, recall that, in my last post, I worried that the resemblance of Newton’s methodology to Bacon’s isn’t enough to establish that Newton was influenced by Bacon’s methodology. If Bacon was just describing a good, general, epistemic method, couldn’t Newton have simply come up with it himself? He was, after all, an exceptional scientist who gave careful thought to his own methodology. Is Ducheyne’s discussion sufficient to establish influence? What do you think?
Crucial Instances in the Principia
Kirsten Walsh writes…
In the General Scholium, which concluded later editions of Principia, Newton described the work as ‘experimental philosophy’:
In this experimental philosophy, propositions are deduced from 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.
On this blog, I have argued that we should take this statement at face value. In support, I have emphasised similarities between Newton’s work in optics and mechanics. For example, I have considered the kind of evidence provided in each work, arguing that both the Principia’s ‘phenomena’ and the Opticks’s ‘experiments’ are idealisations based on observation, and that they perform the same function: isolating explananda. I have also emphasised Newton’s preoccupation in the Principia with establishing his principles empirically. Finally, I have suggested that this concern with experimental philosophy, in combination with his use of mathematics, made Newton’s method unique.
In my last blog post, I wondered if we should regard Newton’s methodology as an extension of the Baconian experimental method, or as something more unique. I have written many blog posts discussing the Baconian aspects of Newton’s optical work (for example, here, here and here), but the Baconian aspects of the Principia are less well-established. I can identify at least three possible candidates for Baconianism in the Principia. The first, suggested by Daniel Schwartz in recent conversation, is that book 3 contains what might be interpreted as Baconian ‘crucial instances’. The second, discussed by Steffen Ducheyne, is that Newton’s argument for universal gravitation resembles Bacon’s method of induction. The third, discussed by Mary Domski, is that the mathematical method employed in the Principia should be viewed as part of the seventeenth-century Baconian tradition. In this post, I’ll focus on Schwartz’s suggestion—the possibility there is a crucial instance in book 3 of the Principia—I’ll address the rest in future posts.
To begin, what is a ‘crucial instance’? For Bacon, crucial instances (instantiae crucis) were a subset of ‘instances with special powers’ (ISPs). When constructing a Baconian natural history, ISPs were experiments, procedures, and instruments that were held to be particularly informative or illuminative of aspects of the inquiry. These served a variety of purposes. Some functioned as ‘core experiments’, introduced at the very beginning of a natural history, and serving as the basis for further experiments. Others played a role later in the process. This included experiments that were supposed to be especially representative of a certain class of experiments, tools and experimental procedures that provided interesting investigative shortcuts, and model examples that came close to providing theoretical generalisations.
Crucial instances are part of a subset of ISPs that were supposed to aid the intellect by “warning against false forms or causes”. When two possible explanations seemed equally good, then the crucial instance was employed to decide between them. To this end, it performed two functions: the negative function was to eliminate all possible explanations except the correct one; the positive function was to affirm the correct explanation.
According to Claudia Dumitru, Bacon’s crucial instances have a clear structure:
- Specify the explanandum;
- Consider the competing explanations (these are assumed to exhaust the possibilities);
- Derive a consequence from one explanation that is incompatible with the other explanation(s);
- Test that consequence.
Are there any arguments in the Principia that look like crucial instances? I think there’s at least one: Newton’s famous ‘Moon test’. Let’s have a look at it.
In proposition 4 book 3, Newton used his Moon test to establish that “The moon gravitates toward the earth and by the force of gravity is always drawn back from rectilinear motion and kept in its orbit”. Here, Newton argued that the inverse-square centripetal force, keeping the moon in orbit around the Earth, is the same force that, say, makes an apple fall to the ground, namely, gravity. I think we can tease out the features of a Baconian crucial instance from Newton’s reasoning here.
Firstly, there is an explanandum: what kind of force keeps the Moon in its orbit and prevents it from flying off into space? Secondly, two possible explanations are provided: the force is either (a) the same force that that acts on terrestrial objects, namely, gravity; or (b) a different force. Thirdly, we have a consequence of (a) that is incompatible with (b): if the moon were deprived of rectilinear motion, and allowed to fall towards Earth, it would begin falling at the rate of 15 1/12 Paris feet in the space of one minute, accelerating so that at the Earth’s surface it would fall 15 1/12 Paris feet in a second. Finally, we see a test of that consequence: the calculations based on the size and motion of the Moon, and its distance from the Earth. The results are taken to support (a) and refute (b).
I have three concluding remarks to make.
Firstly, interpreting the Moon test as a crucial instance involves ‘rational reconstruction’. In the text, Newton starts by calculating the rate at which the Moon would fall, and shows that this supports proposition 4. But I think my reading of this as a crucial instance is supported by Newton’s concluding remarks:
For if gravity were different from this force, then bodies making for the earth by both forces acting together would descend twice as fast, and in the space of one second would by falling describe 301/6 Paris feet, entirely contrary to experience.
Here, Newton described the Moon test as a crucial instance: he used an observation to choose between two competing explanations of the explanandum.
Secondly, when looking for crucial instances in the Principia, it might be tempting to start with the phenomena, listed at the beginning of book 3. Elsewhere, I have argued that these resemble Newton’s experiments in the Opticks, which function as instances with special powers. But the label ‘crucial instance’ describes the function, not the content, of an empirical claim. And so, to see if they provide crucial instances, we need to consider how the phenomena are used. In fact, I think they do provide crucial instances for Newton’s rejection of Cartesian vortex theory in favour of universal gravitation, found at the end of book 2. But again, this requires rational reconstruction.
Finally, there is the issue of historical influence. I have shown that Newton employed the Moon test to decide between two competing explanations, and that this argument resembles one of Bacon’s crucial instances. However, one might think that this was simply a good approach to empirical support, and that Newton was using his common-sense. So perhaps we shouldn’t take this to indicate (direct or indirect) influence. And so I have a question for our readers: was this style of reasoning uniquely Baconian?
‘Epistemic amplification’ and Newton’s laws
Kirsten Walsh writes…
In my last post, I considered the experimental support Newton offers for his laws of motion. In the scholium to the laws, Newton argues that his laws of motion are certainly true. However, in support he only cites a handful of experiments and the agreement of other mathematicians. I suggested that the experiments discussed do support his laws, but only in limited cases. This justifies their application in Newton’s mathematical theory, but does not justify Newton’s claims to certainty. In this post, I will speculate that the laws of motion were in fact better established than Newton’s discussion suggests. I introduce the notion ‘epistemic amplification’ – suggesting that Newton’s laws gain epistemic status by virtue of their relationship to the propositions they entail. That is, by reasoning mathematically from axioms to theorems, the axioms obtained higher epistemic status, and so the reasoning process effectively amplified the epistemic status of the axioms.
I am not arguing that epistemic amplification captures Newton’s thinking. In fact, Newton explicitly stated that epistemic gain was not possible. For him, the best one could achieve was avoiding epistemic loss. (I have discussed Newton’s aims of certainty and avoiding epistemic loss here and here.) I suggest that, objectively speaking, the epistemic status of Newton’s laws increases over the course of the Principia.
To begin, recall that Newton has two projects in relation to the laws:
- The specification of the laws as the axioms of a mathematical system; and
- The justification of laws as first principles in natural philosophy.
Let’s consider the first project. In addition to the support of mathematicians and the experiments that Newton cites, it is plausible that the epistemic status of the laws increases by virtue of their success in the mathematical system: in particular, by entailing Keplerian motion. Kepler’s rules and Newton’s laws of motion have independent evidence: as we have seen, Newton’s laws are weakly established by localised experiments and the ‘agreement of mathematicians’; Kepler’s rules are established by observed planetary motion and were widely accepted by astronomers prior to the Principia. Newton’s laws entail Kepler’s rules, which boosts Newton’s justification for his laws. Moreover, Newton’s laws provide additional support for Kepler’s rules, by telling us about the forces required to produce such motions. The likelihood of the two theories is coupled: evidence for one carries over to the other. So Newton’s laws also boost the justification for Kepler’s rules. Thus, Newton achieves epistemic gain: the epistemic status of the laws, qua mathematical axioms, has increased by virtue of their relationship to Kepler’s rules.
Now let’s consider the second project – the application of the laws to natural philosophy. Again, the discussion in the scholium justifies their use, but not their certainty. I now suggest that these laws, as physical principles, gain epistemic status through confirmation of Newton’s theory. This occurs in book 3, when Newton explicitly applies his mathematical theory to natural phenomena. As I have previously discussed, the phenomena (i.e. the motions of the planets and their moons) are employed as premises in Newton’s argument for universal gravitation. However, the phenomena also support the application of the mathematical theory to the physical world: they show that the planets and their moons move in ways that approximate Keplerian motion. As we saw above, the laws of motion entail Kepler’s rules. So, since the phenomena support Kepler’s rules, they also support the laws of motion. So this is a straightforward case of theory-confirmation.
There is also scope for theory-testing in book 1. Each time Newton introduces a new factor (e.g. an extra body, or a resisting medium), the mathematical theory is tested. For instance, the contrasting versions of the harmonic rule in one-body and two-body model systems provides a test: it allows the phenomena to empirically decide between two theories, one involving singly-directed central forces, the other involving mutually-interactive central forces. Similarly, the contrasting two-body and three-body mathematical systems provide a test: they allow the phenomena to select between a theory involving pair-wise interactions and a theory involving universal mutual interaction. Moreover, in the final section of book 2, Newton shows that, unlike his theory, Cartesian vortex theory does not predict Keplerian motion. Thus, the phenomena seem to support his theory, and by extension the laws of motion, and to refute the theory of vortices. Again, the laws seem to gain support by virtue of their relationship to the propositions they entail.
To summarise, Newton claims that his laws are certainly true, but the support he gives is insufficient. Here, I have sketched an account in which Newton’s laws gain epistemic status by virtue of their relationship to the propositions they entail. ‘Epistemic amplification’ is certainly not something which Newton himself would have had truck with, but the term does seem to capture the support actually acquired by Newton’s laws in the Principia. What do you think?
Are Newton’s Laws Experimentally Confirmed?
Kirsten Walsh writes…
Previously on this blog, I have argued that the combination of mathematics, experiment and certainty are an enduring feature of Newton’s methodology. I have also highlighted the epistemic tension between experiment and mathematical certainty found in Newton’s work. Today I shall examine this in relation to Newton’s ‘axioms or laws of motion’.
In the scholium to the laws, Newton argues that his laws of motion are certainly true. In support, however, he cites a handful of experiments and the agreement of other mathematicians: surprisingly weak justification for such strong claims! In this post, I show how Newton’s appeals to experiment justify the axioms’ inclusion in his system, but not with the certainty he claims.
Newton begins:
- “The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds.”
Newton expands on this claim, discussing firstly, Galileo’s work on the descent of heavy bodies and the motion of projectiles, and secondly, the work conducted by Wren, Wallis and Huygens on the rules of collision and reflection of bodies. He argues that:
- The laws and their corollaries have been accepted by mathematicians such as Galileo, Wren, Wallis and Huygens (the latter three were “easily the foremost geometers of the previous generation”);
- The laws and their corollaries have been invoked to establish several theories involving the motions of bodies; and
- The theories established in (2) have been confirmed by the experiments of Galileo and Wren (which, in turn confirms the truth of the laws).
These claims show us that Newton regards his laws as well-established empirical propositions. However, Newton recognises that the experiments alone are not sufficient to establish the truth of the laws. After all, the theories apply exactly only in ideal situations, i.e. situations involving perfectly hard bodies in a vacuum. So Newton describes supplementary experiments that demonstrate that, once we control for air resistance and degree of elasticity, the rules for collisions hold. He concludes:
- “And in this manner the third law of motion – insofar as it relates to impacts and reflections – is proved by this theory [i.e. the rules of collisions], which plainly agrees with experiments.”
This passage suggests that the rules of collisions support a limited version of law 3, “to any action there is always an opposite and equal reaction”, and that the rules themselves appear to hold under experimental conditions. However, this doesn’t show that law 3 is universal: which Newton needs to establish universal gravitation. This argument is made by showing how the principle may be extended to other cases.
Firstly, Newton extends law 3 to cases of attraction. He considers a thought experiment in which two bodies attract one another to different degrees. Newton argues that if law 3 does not hold between these bodies the system will constantly accelerate without any external cause, in violation of law 1, which is a statement of the principle of inertia. Therefore, law 3 must hold. As the principle of inertia was already accepted, this supports the application of law 3 to attraction.
Newton then demonstrates law 3’s application to various machines. For example, he argues that two bodies suspended from opposite ends of a balance have equal downward force if their respective weights are inversely proportional to the distances between the axis of the balance and the points at which they are suspended. And he argues that a body, suspended on a pulley, is held in place by a downward force which is equal to the downward force exerted by the body. Newton explains that:
- “By these examples I wished only to show the wide range and the certainty of the third law of motion.”
What these examples in fact show is the explanatory power of the laws of motion – particularly law 3 – in natural philosophy. Starting with collision, which everyone accepts, Newton expands on his cases to show how law 3 explains many different physical situations. Why wouldn’t a magnet and an iron floating side-by-side float off together at an increasing speed? Because, by law 3, as the magnet attracts the iron, so the iron attracts the magnet, causing them to press against one another. Why do weights on a balance sometimes achieve equilibrium? Because, by law 3, the downward force at one end of the balance is equal to the upward force at the other end of the balance. These examples demonstrate law 3’s explanatory breadth. But these examples do not give us a compelling reason to think that law 3 should be extended to gravitational attraction (which seems to require some kind of action, or attraction, at a distance).
Newton, clearly, is convinced of the strength of his laws of motion. But this informal, discussion of the experiments he appeals to shows that he ought not be so convinced. As I see it, Newton has two projects in relation to his laws:
1) The specification of the laws as the axioms of a mathematical system; and
2) The justification of laws as first principles in natural philosophy.
I suggest that the experiments discussed give strong support for the laws in limited cases. This justifies their application in Newton’s mathematical model, but it does not justify Newton’s claims to certainty. In modern Bayesian terms, we might say that Newton’s laws have high subjective priors. In my next post, I shall sketch an account in which Newton’s laws gain epistemic status by virtue of their relationship to the propositions they entail.
Cotes’ Preface and Experiment
Peter Anstey writes…
In my last post I introduced Roger Cotes’ famous Preface to the second edition of Newton’s Principia in order to show its importance as an expression of a commitment to experimental philosophy. In that post I focused on Cotes’ critique of the Cartesian vortex theory and the manner in which this attack on the archetypal speculative philosophy formed the bookends of the Principia. In this post I will discuss the role of experiment in Cotes’ comments on experimental philosophy.
The Preface is actually quite a complex essay that has both polemical and expository agendas. On the one hand, Cotes uses it to give a summary of the main theses of the Principia centred around Newton’s theory of gravity. On the other hand, Cotes uses it to defend the theory of gravity against the charge that it is an occult quality, to defend Newton’s system of the world against the Cartesian vortex theory, and to defend the methodology of the work against rival approaches.
On this latter point, Cotes begins by claiming that Newton’s method is ‘based upon experiment’ (The Principia, eds I.B. Cohen and A. Whitman, Berkeley: University of California Press, 1999, p. 386). One might expect here that Cotes will give a list of the sorts of experimental results that Newton achieved or some reference to crucial experiments, but instead he introduces another set of methodological notions: phenomena, principles, hypotheses, analysis and synthesis. It is only later when appealing to various laws, principles and axioms in his summary of Newton’s system of the world that Cotes refers to experiments.
Here is a summary of Cotes’ account of the method of the Principia. Natural philosophy attempts to derive the causes of all things from the simplest of principles and not from contrived hypotheses. These principles are derived from the phenomena by a two-step process of analysis and synthesis. From select phenomena the forces and simpler laws of these forces are ‘deduced’ by analysis. Then by synthesis ‘the constitution of the rest of the phenomena’ is given. In the case of the Principia the relevant force is gravitational attraction and the relevant law is the inverse square law. Though Cotes throws in the laws of planetary motion claiming that ‘it is reasonable to accept something that can be found by mathematics and proved with the greatest certainty’ (p. 389). He also claims, after presenting a summary of the system of the world, ‘the preceding conclusions are based upon an axiom which is accepted by every philosopher, namely, that effects of the same kind –– that is, effects whose known properties are the same –– have the same causes, and their properties which are not yet known are also the same’. Indeed, ‘all philosophy is based on this rule’ (p. 391).
Where then do experiments fit in this picture? The first mention of experiments is in relation to the law of fall. Cotes refers here to pendulum experiments and to Boyle’s air-pump. Next, Huygens’ pendulum experiments are referred to in the discussion of the determination of the centripetal force of the moon towards the centre of the Earth (p. 389). They then appear in the elaboration of the ‘same effect, same cause’ axiom and its application to the attribution of gravity to all matter. Cotes says ‘[t]he constitution of individual things can be found by observations and experiments’ and from these we make universal judgments (p. 391). Thus, ‘since all terrestrial and celestial bodies on which we can make experiments or observations are heavy, it must be acknowledged without exception that gravity belongs to all bodies universally. … extension, mobility, and impenetrability of bodies are known only through experiments’ and so too is gravity. Finally, in recapping the Newtonian method near the conclusion of the Preface Cotes repeats that ‘honest and fair judges will approve the best method of natural philosophy, which is based on experiments and observations’ (p. 398).
What are we to make of the role of experiments here? First, notice how experiments are appealed to in the establishment of laws and the ‘same effect, same cause’ axiom. Second, it is worth pointing out that the ‘same effect, same cause’ axiom is Newton’s second rule of philosophizing: indeed, Cotes uses the very same example as Newton, namely, the falling of stones in America and Europe (see p. 795). Third, notice how without any explanation Cotes extends experiments to experiments and observations. He begins by saying that there are those ‘whose natural philosophy is based on experiment’ and he ends by saying that ‘the best method of natural philosophy, … is based on experiments and observations’. This is not an equivalent expression and while it is consistent with many other methodological statements by experimental philosophers, it still calls out for explanation.
Has Cotes really given an adequate summary of the method of experimental philosophy and has he captured the manner in which experiments are used in Newton’s reasoning in the Principia? In my view he has not. I’d be interested to hear your views?
Newton’s ‘Phenomena’
Kirsten Walsh writes…
On this blog, I have often argued that Newton’s Principia should be characterised as a work of experimental philosophy (for example, here, here and here). To support this argument, I have tended to emphasise similarities between Newton’s work in optics and mechanics. Recently, however, I have noted that some aspects of Newton’s methodology varied according to context. For example, in the Opticks, Newton employed ‘experiments’, but in the Principia, he employed ‘phenomena’. Given that experimental philosophy emphasises observation- and experiment-based knowledge, it is important for my project that I understand Newton’s use of phenomena, and its relationship to observation. In this post, I’ll discuss the phenomena in Principia, and in my next, I’ll discuss the relationship between phenomena and experiments in more detail.
Firstly, let’s consider the origin of the phenomena of Principia. In the first edition of Principia (1687), book 3 contained nine hypotheses. But in the second edition (1713), Newton re-structured book 3 so that it contained only two hypotheses. Five of the old hypotheses were re-labelled ‘phenomena’, and he added one more (phenomenon 2), to bring the total to six:
Phenomenon 1: The circumjovial planets, by radii drawn to the centre of Jupiter, describe areas proportional to the times, and their periodic times – the fixed stars being at rest – are as the 3/2 powers of their distances from that centre.
Phenomenon 2: The circumsaturnian planets, by radii drawn to the centre of Saturn, describe areas proportional to the times, and their periodic times – the fixed stars being at rest – are as the 3/2 powers of their distances from that centre.
Phenomenon 3: The orbits of the five primary planets – Mercury, Venus, Mars, Jupiter, and Saturn – encircle the sun.
Phenomenon 4: The periodic times of the five primary planets and of either the sun about the earth or the earth about the sun – the fixed stars being at rest – are as the 3/2 powers of their mean distances from the sun.
Phenomenon 5: The primary planets, by radii drawn to the earth, describe areas in no way proportional to the times but, by radii drawn to the sun, traverse areas proportional to the times.
Phenomenon 6: The moon, by a radius drawn to the centre of the earth, describes areas proportional to the times.
There are several things to notice about these phenomena. Firstly, they are distinct from data, in that they describe general patterns of motion, rather than measurements of the positions of planetary bodies at particular times. So, while the phenomena are detected and supported by astronomical observations, they are not observed or perceived directly.
Secondly, they are distinct from noumena (or the nature or essence of things), in that they are facts inferred from the observable, measurable properties of the world. They describe the motions, sizes and locations of bodies, but not the substance or causes of these properties of bodies.
Thirdly, they describe relative motions of bodies. That is, in each case, the orbit is described around a fixed point. For example, phenomenon 1 describes the motions of the satellites of Jupiter around Jupiter, which is taken as a stationary body for the purposes of this proposition. In phenomena 4 and 5, the motion of Jupiter is described around the sun, which is taken as stationary.
Fourthly, these phenomena do not prioritise the observer. Rather, each motion is described from the ideal standpoint of the centre of the relevant system: the satellites of Jupiter and Saturn are described from the standpoints of Jupiter and Saturn respectively, the primary planets are described from the standpoint of the sun, and the moon is described from the standpoint of the Earth. And because Newton doesn’t prioritise the observer, effects such the phases and retrograde motions of the planets are not phenomena but only evidence of phenomena.
The re-labelling of these propositions as ‘phenomena’ is somewhat puzzling. The term ‘phenomenon’ has a variety of uses, such as:*
- A particular (kind of) fact, occurrence, or change, which is perceived or observed, the cause or explanation of which is in question;
- An immediate object of sensation or perception (often as distinguished from a real thing or substance); or
- An exceptional or unaccountable thing, fact or occurrence.
But, as we’ve seen, Newton’s ‘phenomena’ don’t properly fit any of these definitions. Can any reader shed light on what Newton really meant by the term?
* Definitions (a) and (c) feature in both C18th and C21st dictionaries, but in the C21st, definition (b) has become more prominent, particularly in philosophy.
UPDATE: I have written a follow-up post.
The Experimental Role of Hypotheses in Newton’s Principia
Kirsten Walsh writes…
In the first edition of Principia (1687), book 3 contained nine hypotheses. But in the second edition (1713), Newton re-structured book 3 so that it contained only two hypotheses. All but one of the old hypotheses were simply re-labelled. Those that specified explanatory constraints were called ‘Rules of Reasoning’, those that were simply unsupported generalisations were called ‘Phenomena’, and only the assumptions about nature were called ‘Hypotheses’.
| 1st edition | 2nd edition |
| Hypothesis 1 | Rule 1 |
| Hypothesis 2 | Rule 2 |
| Hypothesis 3 | Replaced by Rule 3 |
| Hypothesis 4 | Hypothesis 1 |
| Hypothesis 5 | Phenomenon 1 |
| Hypothesis 6 | Phenomenon 3 |
| Hypothesis 7 | Phenomenon 4 |
| Hypothesis 8 | Phenomenon 5 |
| Hypothesis 9 | Phenomenon 6 |
| Lemma 3 | Hypothesis 2 |
Table: Book 3 changes from 1st to 2nd editions.
These changes seem to indicate that Newton’s attitude to hypotheses changed dramatically between 1687 and 1713 – probably in response to Leibnizian criticisms and Cotes’ editorial comments.
On this blog, I have given you many reasons to suppose that, as early as 1672, Newton was working with a clear epistemic distinction between theories and hypotheses. More recently, I have argued that in fact, Newton was working with a three-way epistemic distinction between theories, which are certain and experimentally confirmed, hypotheses, which are uncertain and speculative, and queries, which are not certain, but provide the proper means to establish the certainty of theories. I call this division Newton’s ‘epistemic triad’. I argued that hypotheses perform a distinctive and vital supporting role to theories and queries, and that this role is an enduring feature of Newton’s methodology.
Today I’ll compare the roles of hypotheses and rules of reasoning in book 3, and argue that Newton’s attitude to hypotheses c.1713 was a refinement of his attitude c.1687, but not a dramatic change.
To begin, consider hypothesis 1 (2nd edition): “The centre of the system of the world is at rest.”
Upon introducing this hypothesis, Newton explained that:
- “No one doubts this, although some argue that the earth, others that the sun, is at rest in the centre of the system. Let us see what follows from this hypothesis.”
This is a simplifying assumption. From this hypothesis, in conjunction with Corollary 4 of the Laws of Motion,
- “The common centre of gravity of two or more bodies does not change its state whether of motion or of rest as a result of the actions of the bodies upon one another; and therefore the common centre of gravity of all bodies acting upon one another (excluding external actions and impediments) either is at rest or moves uniformly straight forward”,
Newton derived Proposition 11: “The common centre of gravity of the earth, the sun, and all the planets is at rest.”
This enabled him to calculate the motions of the planets – that is, to deduce the observational consequences of his theory. I consider this to be the chief role of hypotheses, and it is experimental.
Now compare this with how Newton uses his rules of reasoning.
Rule 1 states: “No more causes of natural things should be admitted than are both true and sufficient to explain their phenomena.”
And Rule 2 states: “Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same.”
In his discussion of Proposition 4, Newton explained:
- “And therefore that force by which the moon is kept in its orbit, in descending from the moon’s orbit to the surface of the earth, comes out equal to the force of gravity here on earth, and so (by rules 1 and 2) is that very force which we generally call gravity.”
These rules didn’t help Newton to deduce the observational consequences of his theory in the same way as his hypothesis 1. They provide an important supporting role for his theory, but it is not an experimental role.
To the second edition, Newton also added the General Scholium, in which he (in)famously declared “hypotheses non fingo”. Recently I argued that, in this passage, Newton was not railing against hypotheses in general, but rather, against the use of ‘causal hypotheses’ to illustrate more abstract theories. Given that the first edition did not contain any causal hypotheses, I consider this addition to indicate Newton’s increasing conviction in his method, rather than any dramatic change.
So finally, to summarise the developments between the first and second editions of Principia:
In the first edition:
- Hypotheses are temporarily assumed, untestable propositions that provide a supportive role; and
- There are no causal hypotheses.
In the second edition:
- Hypotheses are temporarily assumed, untestable propositions that provide a supportive experimental role;
- Other temporarily assumed, untestable propositions are given other labels to distinguish them from hypotheses; and
- Emphatically, there are no causal hypotheses.
I see these changes as developments of Newton’s epistemic triad, rather than dramatic methodological changes.
Newton’s 4th Rule for Natural Philosophy
Kirsten Walsh writes…
In book three of the 3rd edition of Principia, Newton added a 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, until yet other phenomena make such propositions either more exact or liable to exceptions.
- This rule should be followed so that arguments based on induction be not be nullified by hypotheses.
Arguably this is the most important of Newton’s four rules, and it certainly sparked a lot of discussion at our departmental seminar last week. Let us see what insights we can glean from it.
Rule 4 breaks down neatly into three parts. I shall address each part in turn.
1. Propositions (acquired from the phenomena by induction) should be regarded as true or very nearly true.
While the term ‘phenomenon’ usually refers to a single occurrence or fact, Newton uses the term to refer to a generalisation from observed physical properties. For example, Phenomenon 1, Book 3:
- The circumjovial planets [or satellites of Jupiter], by radii drawn to the centre of Jupiter; describe areas proportional to the times, and their periodic times – the fixed stars being at rest – are as the 3/2 powers of their distances from that centre.
- This is established from astronomical observations…
Newton uses the term ‘proposition’ in a mathematical sense to mean a formal statement of a theorem or an operation to be completed. Thus, he further identifies propositions as either theorems or problems. Propositions are distinguished from axioms in that propositions are not self-evident. Rather, they are deduced from phenomena (with the help of definitions and axioms) and are demonstrated by experiment. For example, Proposition 1, Theorem 1, Book 3:
- The forces by which the circumjovial planets [or satellites of Jupiter] are continually drawn away from rectilinear motions and are maintained in their respective orbits are directed to the centre of Jupiter and are inversely as the squares of the distances of their places from that centre.
- The first part of the proposition is evident from phen. 1 and from prop. 2 or prop. 3 of book 1, and the second part from phen. 1 and from corol. 6 to prop. 4 of book 1.
Newton appears to be using ‘induction’ in a very loose sense to mean any kind of argument that goes beyond what is stated in the premises. As I noted above, his phenomena are generalisations from a limited number of observed cases, so his natural philosophical reasoning is inductive from the bottom up. Newton recognises that this necessary inductive step introduces the same uncertainty that accompanies any inductive generalisation: the possibility that there is a refuting instance that hasn’t been observed yet.
Despite this necessary uncertainty, in the absence of refuting instances, Newton tells us to regard these propositions as true or very nearly true. It is important to note that he is not telling us that these propositions are true, simply that we should act as though they are. Newton is simply saying that if our best theory fits the available data, then we should regard it as true until proven otherwise.
2. Hypotheses cannot refute or alter those propositions.
In a previous post I argued that, in his early optical papers, Newton was working with a clear distinction between theory and hypothesis. In Principia Newton is working with a similar distinction between propositions and hypotheses. Propositions make claims about observable, measurable physical properties, whereas hypotheses make claims about unobservable, unmeasurable causes or natures of things. Thus, propositions are on epistemically surer footing than hypotheses, because they are grounded on what we can directly experience. When faced with a disagreement between a hypothesis and a proposition, we should modify the hypothesis to fit the proposition, and not vice versa. Newton explains this idea in a letter to Cotes:
- But to admitt of such Hypotheses in opposition to rational Propositions founded upon Phaenomena by Induction is to destroy all arguments taken from Phaenomena by Induction & all Principles founded upon such arguments.
3. New phenomena may refute those propositions by contradicting them, or alter those propositions by making them more precise.
This final point highlights the a posteriori justification of Newton’s theories. In Principia, two methods of testing can be seen. The first involves straightforward prediction-testing. The second is a more sophisticated method, which involves accounting for discrepancies between ideal and actual motions by a series of steps that increase the complexity of the model.
In short, Rule 4 tells us to prioritise propositions over hypotheses, and experiment over speculation. These are familiar and enduring themes in Newton’s work, which reflect his commitment to experimental philosophy. Rule 4 echoes the remarks made by Newton in a letter to Oldenburg almost 54 years earlier, when he wrote:
- …I could wish all objections were suspended, taken from Hypotheses or any other Heads then these two; Of showing the insufficiency of experiments to determin these Queries or prove any other parts of my Theory, by assigning the flaws & defects in my Conclusions drawn from them; Or of producing other Experiments wch directly contradict me…
