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‘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:

  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.

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:

  1. 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”);
  2. The laws and their corollaries have been invoked to establish several theories involving the motions of bodies; and
  3. 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.

Hypotheses and Newton’s Epistemic Triad

Kirsten Walsh writes…

Over the weekend, I participated in a conference on ‘Newton and his Reception’, at Ghent University.  I presented a paper based on my idea that Newton is working with an ‘epistemic triad’.  I had an excellent audience in Ghent, and received some very helpful feedback, but I’d like to hear what you think…

To begin, what is Newton’s ‘epistemic triad’?

In his published work, Newton often makes statements about his purported method in order to justify his scientific claims.  In these methodological statements, he contrasts things that have strong epistemic credentials with things that lack those credentials.  Consider, for example, these passages from his early papers on optics:

    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 … but evinced by ye mediation of experiments concluding directly & wthout any suspicion of doubt. (6 February 1672)
    I shall not mingle conjectures with certainties… (6 February 1672)
    To determine by experiments these & such like Queries wch involve the propounded Theory seems the most proper & direct way to a conclusion. (3 April 1673)

What these passages tell us is that Newton is making a 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 three-way division Newton’s ‘epistemic triad’, and argue that this triad provides the framework for Newton’s methodology.

To support this argument, I defended the following three theses:

Endurance thesis.  There are some general features of Newton’s methodology that don’t change.  These are characterised by the framework of the epistemic triad.

Developmental thesis. There are some particular features of Newton’s methodology that change over time.  These can be characterised as a development of the epistemic triad.

Contextual thesis. There are some particular features of Newton’s methodology that vary with respect to context (namely, mechanics versus optics).  These can be characterised as an adaptation of the epistemic triad to particular contexts.

The developmental and contextual theses are not news to most Newton scholars.  It is commonly accepted that Newton’s methodology changed in important ways over the course of his life, and that there are methodological differences between Principia and Opticks.  The endurance thesis is more problematic, so I made a special effort to show that Newton’s use of hypotheses is more consistent than we think.  I argued that:

  1. In Principia, Newton appears to be working with the same implicit definition of ‘hypothesis’ that he works with in his early optical papers; and
  2. Hypotheses perform similar methodological roles in all of Newton’s natural philosophical work.

I need to do some more work to properly explicate this methodological role.  But, to state it very broadly, Newton temporarily assumes hypotheses, which act as ‘helping premises’ in his inferences from phenomena.  The fact that a statement may appear in Newton’s writing as a hypothesis, and then reappear later in a query, rule of reasoning, or phenomenon, has convinced many Newton scholars that Newton is inconsistent in his use of hypotheses. Against this conviction, I argue that Newton applies the label ‘hypothesis’ to things that perform a particular function, rather than to a particular claim.

The Aims of Newton’s Natural Philosophy

Kirsten Walsh writes…

In a previous post I discussed the aim of absolute certainty in Newton’s early optical papers.  I argued that this aim provides the link between Newton’s mathematical and experimental methods.  This quest for certainty is an enduring feature of Newton’s natural philosophy, leading to a modest natural philosophical agenda.  For example, in the General Scholium to the Principia (1713), Newton writes:

    “I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses … And it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.”

But is this really enough, for Newton?  Apparently it’s not.  In the very next paragraph, Newton begins to speculate on the “subtle spirit” that permeates bodies and might be operative in various phenomena.  It looks like he is proposing a causal explanation of universal gravitation.  However, these speculations end before they really begin, when Newton concludes that “there is not a sufficient number of experiments to determine and demonstrate accurately the laws governing the actions of this spirit.”

This is the final line of Principia.  And, for such a controversial book, this is a rather inauspicious ending.  But I think we can glean something about the aims of Newton’s natural philosophy from this.

To begin, we need to distinguish between what Newton wants to achieve, and what he thinks he can achieve.  Newton wants to give a complete, true theory of the world – including an account of the motions of the planets, the cause of gravity, and even God’s relation to the natural world.  But, in the trade-off between completeness and truth, Newton sides with truth.  For, as he writes in an unpublished Preface to Principia (mid-1710s), “still it is better to add something to our knowledge day by day than to fill up men’s minds in advance with the preconceptions of hypotheses.”

Newton’s modesty and restraint should not be misinterpreted as lack of epistemic ambition.  The surest way to achieve absolute certainty would be to keep his domain of inquiry as narrow as possible.  But Newton doesn’t do this.  Instead, he pushes at the boundaries of what can be known with certainty.  This is demonstrated by his use of Baconian Induction to make increasingly general claims about gravity.  Newton ambitiously generalises from pendulums, to terrestrial bodies, to all bodies.  In an unpublished Preface to Principia, he writes:

    “But it has also been shown in the Principia that the precession of the equinoxes and the ebb and flow of the sea and the unequal motions of the moon and the orbits of comets and the perturbation of the orbit of Saturn by its gravity toward Jupiter follow from the same principles and what follows from these principles plainly agrees with the phenomena.”

So what do those final two paragraphs of the General Scholium tell us about the aims of Newton’s natural philosophy?  I. Bernard Cohen says that the General Scholium is similar to the discussions that are found in scientific papers today: Newton is discussing the implications of his results and suggesting areas of further research.  On this reading, Newton is saying that there are two jobs ahead:

  1. To give a causal explanation of gravity; and
  2. To apply the theory of gravity to other phenomena in order to solve other problems.

Importantly, Newton thinks that we can begin on (2) without waiting to complete (1).  This is why Newton says it is enough that he has established that gravity exists and acts according to certain laws.

Related Posts: Newton on Certainty, Newton’s 4th Rule for Natural Philosophy.

Should we call Newton a ‘Structural Realist’?

Kirsten Walsh writes…

At our symposium last week, someone wondered if we can characterise Newton as a ‘structural realist’.  It is certainly anachronistic to attempt to interpret Newton’s epistemic stance in light of the present-day scientific realism debate.  But the sin of anachronism may be forgiven, if it advances our understanding.  So let us see what advantages this interpretation may provide.

Briefly, structural realism is the view that epistemically, a scientist should only commit herself to the mathematical or structural content of her theories, and remain sceptical about the unobservable entities posited by those theories.

To characterise Newton as a structural realist, one might make the following argument:

    P1. Newton is a realist about his theories, but not about his hypotheses.
    P2. Newton’s theories make claims about theoretical structures, whereas his hypotheses make claims about unobservable theoretical entities.
    C. Therefore, Newton is a realist about theoretical structures, but not about unobservable theoretical entities.

Firstly, consider Newton’s hypothesis/theory distinction.  In a previous post I argued that Newton claims that his doctrine of light and colours is a theory, not a hypothesis, for three reasons:

    T1. It is certainly true, because it is supported by (or deduced from) experiment;
    T2. It concerns the physical properties of light, rather than the nature of light; and
    T3. It has testable consequences.

In contrast, he attaches no special epistemic merit to his corpuscular hypothesis because:

    H1. It is not certainly true, because it is not supported by experiment;
    H2. It concerns the nature of light; and
    H3. It has no testable consequences.

T1 and H1 support P1.  They tell us that Newton is a realist about theories because they can be shown to be true on the basis of experiment.  Moreover, he is not a realist about hypotheses because they cannot be shown to be true on the basis of experiment.  This highlights an important feature of Newton’s methodology: Newton is only epistemically committed to those things that are demonstrated experimentally.

T2 and H2 appear to support P2, but only if the ‘entity/structure’ distinction maps onto Newton’s ‘nature/physical properties’ distinction.  Prima facie, it does.  While Newton probably wouldn’t have been comfortable with the entity/structure distinction, the structural realist debate is often framed in terms of the nature/physical properties distinction.  For example, here’s how the Stanford Encyclopedia of Philosophy describes the structural realist position:

    Structural realism is often characterised as the view that scientific theories tell us only about the form or structure of the unobservable world and not about its nature.  This leaves open the question as to whether the natures of things are posited to be unknowable for some reason or eliminated altogether.

So it looks like the argument for characterising Newton as a structural realist is well-supported by Newton’s distinction between theory and hypothesis.  But what do we gain by characterising Newton in this way?

Chris Smeenk recently pointed out to me in an email that the structural realist label identifies a distinctive feature of Newton’s methodology.  Namely, that he is epistemically committed to his abstract mathematical structures.  He is not an instrumentalist about his theories, but neither is he a realist about the nature of the phenomena they describe.  This might shed some light on the optical debate of the early 1670s, for unlike his contemporaries, Newton does not think there is a contradiction in believing that his theory of light is true, while not committing himself to any particular doctrine regarding the nature of light.

Is this a large enough pay-off to warrant the offence of anachronism?  What do you think?

In this brief post, I have only considered Newton’s attitudes to his own theories.  There are other questions to be raised in connection with structural realism, for example, is Newton a structural realist about the history of science?  In other words, what is Newton’s epistemic commitment to the theories of his predecessors?  I shall leave this question for another time.

On another note, we were very pleased with how last week’s symposium went.  We look forward to telling you all about it next Monday.

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.