**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.