©The Royal Society/Richard Valencia.

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…

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