## Wednesday, January 30, 2013

### Whewell on Newton's Laws IV: The Second and Third Laws

This accidentally got stuck in the draft stage. It's imperfect -- it really needs to address the historical questions better, for instance, and I didn't finish putting in all relevant citations; but I publish it just to get it out of draft so I can go on to other things and come back to this later when I am able.

It will be convenient in discussing the Second and Third Laws of Motion to proceed somewhat more schematically than with the First Law, keeping in mind what was said about the First Law as being a product of both rigorous conceptual clarification of the Idea of Cause and experimental specification, thus making it in one sense necessary and in another sense contingent and revisable. Mutatis mutandis, this is true for the Second and Third Laws as well, although their histories and philosophical implications (understood as causal laws in Whewell's sense) are somewhat more elaborate.

Second Law (as found in Newton): Change of motion is proportional to the impressed motive force, and occurs in a straight line with respect to that impressed force.
General Causal Axiom: Causes are measured by their effects. (Whewell argues that this causal axiom presupposes certain conditions in order to apply, namely, those that make possible additive measurement in the first place.)
Causal Axiom Specified to Forces: The accelerating quantity of a force is measured by the acceleration produced.
Primary Experimental Component: The velocity and direction which a body already possesses are not, either of them, causes which change the acceleration produced.

The history of the Second Law as Whewell understands it is closely bound up with the history of both early modern astronomy and increasing sophistication in the understanding of gravity in the early modern period. If we start with accelerations, changes of velocity, and try to explain them, we know that if our acceleration has been caused by an impressed force, what can be said about this impressed force has to be determined from the acceleration, its effect; and that, since effects and causes are proportionate to each other, we can give a measure to the cause based on the effect, and we can therefore give a measurement to the force based on the acceleration. But in so doing, we have to rule out interfering factors. This is actually quite difficult to do, since possible interfering factors are often found even in our ordinary experience. The case of gravity, however, appears to give us a cleaner case of force causing acceleration than we usually find, in which the interfering factors can be reasonably minimized. The key issue is that we can reasonably assume -- because the assumption is thoroughly in conformity with our experience -- that gravity is a constant cause. If the earth rotates, then every point of its surface is in motion. If we drop a stone from the top of a tall building, then, we might expect that the stone would be left behind as the earth continued its rotation and the tower moved away from it. However, if the stone were also subject to the same motion, having the same force working on it, then the movement of the stone relative to the tower would be the same as if they were both at rest. This ties in with considerations of the First Law, which gives the proper relation between rest and force; without something reasonable clear and like the First Law, there are quite a few logically possible interfering factors that we will have difficulty sweeping away. Once people began to use something more like the First Law, however, it could be argued that the circular motion of the earth and the linear motion of the stone did not diminish or augment each other, and various experimental studies of relative motion strongly suggested that the change in the vertical movement of a dropped or thrown stone does not depend on any velocity or direction the stone already has. "We so willingly believe in the simplicity of laws of nature, that the rigorous accuracy of such a law, known to be at least approximately true, was taken for granted, till some ground for expecting the contrary should appear."(PIS volume 1, 229); however, there are other things that had to come together to establish the Second Law. It is necessary, for instance, to consider not just bodies in gravity but also bodies interacting with each other, and on that basis come to recognize that the same kinds of causes that can explain motion can also explain tension. To get the Second Law you need to work your way to seeing how something like it could explain not just relatively pure cases but complicated and non-apparent cases; and despite our calling it a Law of Motion it has to cover static cases as well.

Thus, without getting into all the details Whewell chases down on the historical side in the History of the Inductive Sciences and, in more summary form, the Philosophy of the Inductive Sciences, we can see that the Second Law is highly bound up, and can even be seen as a sort of summary of, much of the history of what we often call the Scientific Revolution. Whewell doesn't quite think this way himself, not because he doesn't think Galileo and the rest were a major leap forward, since he holds that the medieval period was largely stagnant except in practical matters, but because he tends not to think in terms of Science but in terms of Sciences, and to think of scientific progress not in terms of single events but in cycles of varying speeds, Epochs as he calls them, where the discoveries of prior Epochs are put in new light.

When Whewell says (ETM 5), "All causes must be measured by their effects; and force, as a conception included in the idea of a cause, must be measured by the effects which it produces," this gives what he sees as the logical structure -- it explains, so to speak, why the Second Law can explain anything -- but the clarifications and new data required to make this structure clear are far from being as simple and, in fact, make up a massive portion of the modern history of experiment. As he says elsewhere (First Principles of Mechanics v), "When it is seen through how many attempts, and after how many errors of the most intelligent speculators, every one of these doctrines has been reduced to its final simplicity and certainty, it will perhaps be more evident how entirely they depend on experiment for their proof, and how far from easy the discovery was."

Third Law (as found in Newton): To every action is always opposed an equal reaction; or the mutual actions of two bodies on each other are always equal and directed to contrary parts.
General Causal Axiom: In measurable causal interactions, action is always accompanied an equal and opposite reaction.
Causal Axiom Specified to Forces: In the direct mutual action of bodies, the momentum gained and lost in any time is equal. (Compare also D'Alembert's Principle, which is closely related: When any forces produce motion in any connected system of matter, the motive quantities of force gained and lost by the different parts must balance each other according to the connection of the system.)
Primary Experimental Component: The connection of the parts of a body or of a system of bodies, and the action to which a system of bodies is already subject, are not, either of them, causes which produce any new change in the effects of any additional action.

In many ways, the Third Law is the Law of Motion that most stands out. When Euler later wrote the Mechanica, for instance, he derived the First and Second Laws (they are not laws for Euler but theorems derived from more general principles), but there's nothing definitely like the Third Law. Likewise, the history of statics plays a much larger and more direct role in the historical lead-up to it (at least as Whewell understands it) than it does in the other two. In statics it began to be recognized that "When two equal weights are supported on the middle point between them, the pressure on the fulcrum, is equal to the sum of the weights" (PIS II); an analogous rule began to be recognized in hydrostatics. The Third Law is in part a direct generalization of this. This special connection with statics seems to be related to what Whewell thinks of as the experimental component of the Law, which involves determining what is involved in "the connection of the parts of a body". Moreover, the third law is specifically about bodies, and was itself bound up in the development of thought about bodies. Whewell argues this is not accidental; the kind of reaction indicated by the Third Law is "inseparable from our conception of body or matter" (MPCR) -- hence the close historical connection between the concepts of matter and inertia, inertia being a force-term.

The principle is not, however, purely a generalization from statics, since, as Whewell notes, Galileo gives a principle structurally similar to it; he uses it in revised editions of his Dialogues his principle to patch a hole in his reasoning about inclined plans as he had originally presented it. Galileo in turn had been simplifying an assumption that velocities acquired in rolling down inclined planes are of the same height.

Clarification of the Third Law required clarifying the other two laws as well, according to Whewell. As he says:

The mistake of Aristotle and his followers, in maintaining that large bodies fall more quickly than small ones, in exact proportion to their weight, arose from perceiving half of the third law of motion, that the velocity increases with the force which produces it; and from overlooking the remaining half, that a greater force is required for the same velocity, according as the mass is larger. The ancients never attained to any conception of the force which moves and the body which is moved, as distinct elements to be considered when we enquire into the subject of motion, and therefore could not even propose to themselves in a clear manner the questions which the third law of motion answered. [PIS 2, 589]

Thus the Third Law is distinctive in a number of ways. What is perhaps most significant about it, however, is that, as interpreted by Whewell it involves mutual causation (ETM 6):

For the action and the reaction may each be conceived as determining the other; they are mutually cause and effect; and, therefore, depend each upon the other by the same law. And, therefore, there can be no reason why one should be greater or less than the other, or in a different line. They are necessarily equal and opposite.

This has the direct implication that cause and effect are (at least in this case) simultaneous, and that anything to which the Third Law applies is a counterexample to any view of causation that insists that cause must come before the effect. It is certainly true that we commonly appeal to before and after in discussing causation. But if there is any interval of time between the action of force and its reaction, then for that interval of time the Third Law would be false -- an action would have no reaction. Thus the Third Law, interpreted as a causal law, implies simultaneous causation. Why, then, do we often talk about cause and effect as if they were successive? In part, Whewell thinks (he borrows some clarifying distinctions from a reviewer of his Philosophy of the Inductive Sciences), because we are often talking about indirect causation, and indirect causation does not involve mere application of force, but requires time for parts to move so as to create a cumulative effect. If you think of a machine with gears, for instance, every part that direct imposes force on any other part (one gear on another) has an effect immediately and for as long as it presses on the second part. But the second gear's role in the mechanism is not merely to resist the first one; it is set in motion by the first gear, and its movement takes time. Cause and effect are not successive by nature or in themselves; in the case of forces, it is the combination of applied force with the time required for movement that makes us think of effect as something that comes after the cause in time. This can be generalized to other kinds of cause and effect (PIS 2, 644):

...we have in every case a uniform state, or a state which is considered uniform, or at least normal; and which is taken as the indication and measure of time; and we have also change, which is contemplated as a deviation from uniformity, and is taken as the indication and measure of cause.

It is the combination of change and invariance which tangles together our ideas of Time and Cause. But according to Whewell the work done on the Third Law shows at the very least that, within certain experimental bounds, this temporal element is incidental to the questions of cause and effect themselves. Thus the causal interpretation of the Third Law has some teeth: since causation as such does not require a temporal direction, the interpretation is inconsistent with accounts of causation, like Hume's, that assume causes must be temporally prior to their effects.