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# Tag Archives: mathematics

## 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!

## University of Sydney

20 March, 2014

9:15-5:30

Program:

• 9.15 Katherine Dunlop (Texas): ‘Christian Wolff on Newtonianism and Exact Science’
• 10.45 Coffee
• 11.00 Peter Anstey (Sydney): ‘From scientific syllogisms to mathematical certainty’
• 12.30 Lunch
• 2.00 Kirsten Walsh (Otago): ‘Newton’s method’
• 3.30 Stephen Gaukroger (Sydney):  ‘D’Alembert, Euler and mid-18th century rational mechanics: what mechanics does not tell us about the world’
• 5.00 Wind up

Contact:    Prof Peter Anstey

Phone:       61 2 9351 2477

Email:       peter.anstey@sydney.edu.au

RSVP:      Here

## Newton on Experiment and Mathematics

Kirsten Walsh writes…

In my last post, I discussed our 20 revised theses and why I altered thesis 5.  In this post, I’ll discuss why I replaced thesis 8.

In 2011, I claimed that:

8.  The development of Newton’s method from 1672 to 1687 appears to display a shift in emphasis from experiment to mathematics.

But at the start of this year, I replaced this thesis with a new thesis 8:

8.  In his early work, Newton’s use of the terms ‘hypothesis’ and ‘query’ are Baconian.  However, as Newton’s distinctive methodology develops, these terms take on different meanings.

Since my new thesis is a replacement of the original thesis, rather than a modification, two explanations are required.  So in today’s post, I’ll tell you why I decided to remove my original thesis 8, and in my next post, I’ll tell you about my new thesis 8.

I originally included thesis 8 because there are some obvious differences in the styles of Newton’s early work on optics and his Principia.  In Newton’s first paper on optics (1672), there is a strong emphasis on experiment.  Experiment drives his research and guides his rejection of various possible explanations of the phenomena under consideration.  Ultimately, he presents an Experimentum Crucis as proof for the certainty of his proposition that white light is heterogeneous.  In contrast, the Principia (1687) displays a strong emphasis on mathematics.  The full title of the work, the Author’s Preface to the Reader, and the fact that Book I opens with 11 lemmas outlining the mathematical framework of the work are just a few features that make it clear that Principia is primarily a mathematical treatise.

I now think that my original thesis 8 is misleading.

Firstly, as I have emphasised on this blog, Newton’s early work had a mathematical style that made it unique among his contemporaries.  While they recognised him as an experimental philosopher, his claims of obtaining certainty via geometrical proofs set him apart from the Baconian-experimental philosophers.  Moreover, his methodological statements show evidence of a tension between experiment and mathematical certainty.  For example, he says that the science of colours,

“depend[s] as well on Physicall Principles as on Mathematicall Demonstrations: And the absolute certainty of a Science cannot exceed the certainty of its Principles.  Now the evidence by wch I asserted the Propositions of colours is in the next words expressed to be from Experiments & so but Physicall: Whence the Propositions themselves can be esteemed no more then Physicall Principles of a Science.”

Secondly, Newton continued to identify as an experimental philosopher until the end of his life.  For example, in the General Scholium at the end of Principia, he says:

“and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy.”

This resembles Newton’s earlier emphasis on grounding propositions on empirical evidence, rather than on speculative conjectures.

Thirdly, in Principia, Newton appears to be negotiating a similar tension between experiment and mathematical certainty that we saw in his early work.  For example, in the Scholium to the Laws of Motion he asserts the certainty of his Laws, while at the same time, acknowledging their experimental basis:

“The principles I have set forth are accepted by mathematicians and confirmed by experiments of many kinds.”

And:

“By these examples [i.e. the experiments mentioned above] I wished only to show the wide range and the certainty of the third law of motion.”

From these three points, we can see that the methodological differences between Newton’s early papers and Principia aren’t as great as they first appear.  But I did not remove my original thesis 8 because I think that the methodology of the 1672 paper is precisely the same as the methodology displayed in Principia.  Rather, I don’t think my original thesis 8 captures what is important about these differences.

As I have explained here, my project is to distinguish between those features of Newton’s methodology that changed, and those that stayed the same.  Some aspects of Newton’s methodology developed over time.  For example, he came to value geometrical synthesis over algebraic analysis.  Other aspects of his methodology varied according to context.  For example, in Opticks, he employs ‘experiments’ and ‘observations’, but in Principia, he employs ‘phenomena’.  But this triumvirate of methodological ideas – experiment, mathematics and certainty – should be considered an enduring feature of Newton’s methodology.

## Newton and the Case of the Missing Calculus

Kirsten Walsh writes…

The case of the missing calculus is well-known.  Newton (co-)invented calculus in the late 1660s, and he wrote Principia in the late 1680s.  It would be natural to expect that Newton used the calculus in Principia.  But it seems that he didn’t.  Instead, Newton wrote Principia in the style of Euclid’s Elements, that is, using Classical Greek geometry.  This is surprising indeed, given the powerful new tool he had at his disposal.  What should we make of this?

Almost thirty years after the publication of Principia, Newton explained that he had used algebraic calculus to discover the propositions of Principia, but used classical geometry to demonstrate them:

“By the help of the new Analysis [i.e. algebraic calculus] Mr. Newton found out most of the Propositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry.  And this makes it now difficult for unskilful men to see the Analysis by which those Propositions were found out.”

But Newton was lying.  Scholars have found no evidence that he wrote or developed Principia in any other way than the published form.  Moreover, few, if any, of the propositions in Principia can even be presented in the form of algebraic calculus.

This raises two questions:

1. Why did Newton lie?
2. Why did Newton eschew modern algebraic calculus in favour of classical geometry?

These questions have been discussed by numerous scholars including A. Rupert Hall and I. Bernard Cohen.  The answer to (1) can be found in Newton’s priority dispute with Leibniz.  The answer to (2) was summarised neatly by Thony Christie last year:

“Put simply Newton had serious doubts about the reliability of the new analytical mathematics and that is why he didn’t use it for his magnum opus.”

But what caused these doubts?

In 1714, Newton wrote that the algebraic calculus is “arithmetic applied to geometrical matters… Its operations are complicated and excessively susceptible to errors, and can be understood by the learned in algebra alone”.  Whereas geometry “may be appreciated by the great majority and thus most impress the mind with [its] clarity”.  One might wonder why Newton bothered to invent algebraic calculus at all!

Well it seems that Newton wasn’t always so anti-algebra, nor was he always so interested in classical geometry.  In fact, as an undergraduate, Newton didn’t read the ancients.  Rather, he read a few modern summaries of the ancient texts, building his own mathematics on the algebraic work of mathematicians such as Descartes, Wallis and Barrow.

Newton seems to have become interested in classical geometry in the late 1670s, after re-reading Descartes’ La GéométrieLa Géométrie was an attempt to unite algebra and geometry – Descartes aimed to show how symbolic algebra could be applied to the study of plane curves.  Guiccardini writes:

“[Descartes’] tract could be read as a deliberate proof of the superiority of the new analytical method, uniting symbolic algebra and geometry, over the purely geometrical ones of the ancients.”

Newton was very critical of Descartes’ text, writing comments such as “error” and “I hardly approve” in the margins.  He even drafted a paper entitled ‘Errors in Descartes’ Geometry’. To find support for his position, Newton began to read the ancient texts, including Pappus.

Newton wrote:

“To be sure, [the ancients’] method is more elegant by far than the Cartesian one.  For [Descartes] achieved the result by an algebraic calculus which, when transposed into words (following the practice of the Ancients in their writings), would prove to be so tedious and entangled as to provoke nausea, nor might it be understood.  But they accomplished it by certain simple propositions, judging that nothing written in a different style was worthy to be read, and in consequence concealing the analysis by which they found their constructions.”

Newton was neither the first, nor the only, philosopher to equate algebra and geometry with the ancient methods of analysis and synthesis respectively.  But he was the first to reject modern algebraic calculus in favour of ancient geometry.  (If only because he was the first to invent it!)  Does Newton’s rejection of algebraic calculus stem from his anti-Cartesian stance?  What if Newton had never re-read Descartes’ Géométrie?  Could his priority dispute with Leibniz have been avoided?

## Newton on Certainty

Kirsten Walsh writes…

A few weeks ago, I said that in Newton’s early optical papers:

Newton claims that his doctrine of colours is a theory, not a hypothesis, for three reasons:
1.  It is certainly true, because it is supported by (or deduced from) experiment;
2.  It concerns the physical properties of light, rather than the nature of light; and
3.  It has testable consequences.

From this set of criteria, we can see that early-Newton’s strong anti-hypothetical stance is closely related to his goal of generating theories that are certainly true.  Students from Florida have pointed out that Newton’s criterion of certainty seems to set the bar quite high.  Indeed it does.  So today I will explain early-Newton’s goal of absolute certainty and why he thought it was achievable.

For Newton, absolute certainty is closely related to mathematics – he wants to achieve certainty in the science of colours by making it mathematical.  In his first letter to the Royal Society, he says:

A naturalist would scearce expect to see ye science of those become mathematicall, & yet I dare affirm that there is as much certainty in it as in any other part of Opticks.  For what I shall tell concerning them is not an hypothesis but most rigid consequence, not conjectured by barely inferring ’tis thus because not otherwise or because it satisfies all Phænomena (the Philosophers universall Topick,) but evinced by ye mediation of experiments concluding directly & without any suspicion of doubt.

In a letter to Hooke, Newton says, ideally the science of colours will be “Mathematicall & as certain as any part of Optiques”.  However, absolute certainty is difficult to achieve because the science of colours

depend[s] as well on Physicall Principles as on Mathematicall Demonstrations: And the absolute certainty of a Science cannot exceed the certainty of its Principles.

Thus, Newton thinks that absolute certainty is also closely related to experiment.  It is no accident that, in his first paper, Newton attempts to establish the physical principles of colour experimentally by focussing on refrangibility rather than colour of light.  It would have been difficult to measure precisely changes in colour, but Newton was able precisely to measure degrees of refraction and lengths of refracted images.  He hardly even mentions colour until he believes he has established that white light is a mixture of differently refrangible rays.  When he is ready to reveal his theory of colour, he does so by first asserting that there is a one-to-one correspondence between refrangibility and colour of light rays.  Newton claims that he has established the physical principles of colour with absolute certainty.

When he reveals his theory of colour, he does so in a quasi-mathematical style.  In a letter to Oldenburg, Newton says:

I drew up a series of such Expts on designe to reduce ye Theory of colours to Propositions & prove each Proposition from one or more of those Expts by the assistance of common notions set down in the form of Definitions & Axioms in imitation of the Method by which Mathematicians are wont to prove their doctrines.

This quasi-mathematical ‘proof’ of his theory of colours is set out in his reply to Huygens.

To summarise, Newton’s mathematical method and his experimental method are linked by his notion of absolute certainty.  Newton claims his theory of colours is certainly true, because (1) his physical principles are established experimentally and are certainty true, and (2) he can use these physical principles as the basis of his mathematical proof.  That a lengthy and sometimes heated debate followed Newton’s original paper, shows that his opponents weren’t as convinced by his careful demonstration as he was.