IS THERE A BIGGER PAYOFF?
I suggest that this is at least part of what is going on when Newton and Mill propose their methodologies. Norton and the rest of you may agree, but say: “Big deal! We'll give you that.
Induction is still local and not formally warranted.”There is, I think, a good deal more here. Newton and Mill are attempting to present very general rules that constitute their “scientific method” for proving propositions in science. Both explicitly reject the hypotheti- co-deductive method, according to which you prove or confirm a hypothesis by deriving observational predictions from it which are found to be true. What Newton wants instead are what he calls “deductions from the phenomena” that are made general by induction. What Mill wants in cases that involve multiple causal hypotheses is, I think, close to Newton. Mill calls it the “deductive method.” It has three steps: first, a series of causal- inductive arguments to the causal laws in question (this requires the use of Mill's methods to determine causation); second, what Mill calls “ratiocination,” which involves explaining various phenomena and deriving new predictions from the set of causal laws; third, experimental verification of these predictions. Some comments are in order.
1. The set of rules Mill offers—his methods for determining causation and his “deductive method”—is both universal and abstract, perhaps sufficiently so to merit Norton's epithet “formal template.” The same is true, I think, for Newton's rules.
2. Such rules have a bite to them. They are not trivial or uninformative. They tell scientists what to do and not to do in attempting to prove causal hypotheses in a system. And they go against various other such methodologies, including in particular that of Whewell, since, as Mill notes, Whewell omits the first causal-inductive step. On occasion such rules are explicitly cited against scientists who flout them, as in the case of Newton himself criticizing Descartes' physics for flouting induction from observed phenomena, or Lord Brougham at the beginning of the nineteenth century excoriating Thomas Young for defending the wave theory of light on the basis of no Newtonian inductions to the assumptions of the wave theory and no experimentally confirmed predictions,[37] or in our own time criticisms of string theory by physicists citing methodological grounds similar to those of Newton and Mill.[38]
3.
Although these methods are formulated universally and can be expressed in formal terms, they require the use of empirical assumptions to carry out in such a way that the resulting theoretical system is “proved.” The causal-inductive inferences in the first step of Mill's “deductive method” need to be justifiable in the light of what is known, the derivations must be correct, and the experiments and instruments used must be appropriate. So even if step 2 can be accomplished by a priori “calculation,” steps 1 and 3 need to be secured empirically. Admittedly, the set of rules is not “formal” in the way that formal rules of deductive inference are. But simply because the validity of an inference to the truth or probability of the system of hypotheses cannot be decided entirely a priori by reference to formal structure, it does not follow that the formal structure of the inference pattern plays no justificatory or explanatory role.There is analogy here between what Mill is doing in the case of inductive inference and what, in the next century, Hempel (1965) did with the concept of scientific explanation. Hempel formulated a set of conditions, which included both formal ones (he called them “logical”) and material ones (he called them “empirical”) for his two models of explanation: deductive-nomological and inductive-statistical. Both have formal requirements pertaining to types of premises (singular and general), empirical character, and type of logical connection between premises and conclusion. And both have material conditions involving requirements of truth and/or empirical confirmation. For Hempel both sorts of requirements must be satisfied to have a correct or justified scientific explanation. What Mill is claiming for scientific inference is that satisfying the formal requirements of his “deductive method,” which calls for causal-inductive inferences of certain forms, calculation, and derivations of certain types, is a necessary condition for proving the truth of the theoretical system “beyond reasonable doubt.” He is also claiming that satisfying the material requirements of his conditions—i.e., making empirically justified causal-inductive inferences and empirically established predictions—is another necessary condition for proof “beyond reasonable doubt.”
4.
None of this means that in constructing an empirical “proof” of the sort Mill or Newton had in mind one must explicitly invoke or appeal to the three steps outlined by Mill in his “deductive method,” or to the four rules of Newton at the beginning of the third book of his Principia. But the same is true in a deductive proof: in constructing such a proof one need not invoke or appeal to any of the formal principles of deduction that logicians love to classify and explicitly use. One cannot conclude from this that the principles, in either the deductive or the inductive case, have no justificatory force.5. Finally, none of this precludes peaceful coexistence with physically indeterministic systems of the sort John Norton describes. In his dome example a point mass is initially motionless at the apex and can slide at any time in any radial direction without any disturbance at the apex. The physics of the situation as described by Newton's equations of motion admits various solutions in such a way that we can say only what is possible, not what will happen or will probably happen. And this is not a matter of our ignorance, but of the world.
If so, and if, following Newton's own methodology, we agree that Newton's equations of motion are established inductively, then we have a situation in which, using inductively supported laws, we show that we cannot predict with certainty or probability what the motion of the point mass will be. It's not that induction fails for this example. It works very well. It tells us that the particular physical situation is such that it is impossible to predict with certainty or probability when, where, or if the point mass will move; it gives us only possibilities. Of course, if you are a “crazy Bayesy” you will demand a degree of belief, i.e., a probability, for every proposition. Newton was a bit crazy, but neither he, nor Mill, nor John Norton is a Bayesian.
5.