1. Introduction

For some time now, induction, particularly inductive generalization, has been under siege. A frontal attack, based on the sort of skepticism attributed to Hume, is made by Popper (Reference Smolin1959), according to which no inductive generalization can be rationally justified. But other nonskeptical attacks are meant to be just as devastating. For example, it is claimed that good scientists do not use inductive generalization and that if and when they appear to be using such reasoning, they are really employing something else, such as “deductive reasoning” (again, Popper), or “inference-to-the-best-explanation” (Harman Reference Hempel1965; Lipton Reference Mayo1991), or statistical reasoning invoking “severe testing” (Mayo Reference McGuire1996). It is also claimed that inductive generalization, as typically characterized, is too incomplete, too vague, or too puerile to adequately express complex and sophisticated scientific reasoning (Whewell Reference Woit1847/1967). At best, it occurs only when one can directly observe instances of a low-level empirical generalization, and it is not applicable to more theoretical science. Finally, some have claimed that although there is such a thing as inductive generalization, contrary to what inductivists believe, it is not subject to formal and universal rules of the sort usually proposed (Norton Reference Popper2005).

I want to argue that induction, as traditionally formulated by inductivists such as John Stuart Mill and Isaac Newton, can withstand these attacks. Or at least these inductivists can put up a better fight than their opponents might imagine. I will look at a historically important controversy between William Whewell, on the one hand, and Newton and Mill, on the other. Admittedly, all three claim to be endorsing inductive generalization. But Whewell claims that Newton and Mill really got it wrong. The latter are usually thought of as the more typical inductivists, deserving of philosophical scorn, while Whewell is currently more popular with a disparate group who classify him as one of their own, as a hypothetico-deductivist, a defender of inference to the best explanation, a Quinean holist, or even a Kantian a priorist. I propose to defend Mill and Newton against this formidable opponent.Footnote ¹ I will also defend Mill and Newton against another more recent formidable opponent, John Norton, who goes even further than Whewell in his rejection of inductivists such as Mill and Newton.

Mill (Reference Newton, Cohen and Whitman1872, 188) offers a classic definition of induction: “Induction, then, is that operation of the mind by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, Induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times.”Footnote ²

Isaac Newton never explicitly defines “induction” in his works but, like Mill, considers it to be a necessary component of scientific reasoning to general propositions: “In … experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies, and the laws of motion and the law of gravity have been found by this method” (Newton Reference Norton and Achinstein1999, 943). The closest Newton comes to a definition of induction is in rules 3 and 4 in the opening section of book 3 of the Principia:

Newton (Reference Norton and Achinstein1999, 795) treats rule 3 as an inductive rule applicable to “qualities of bodies that cannot be intended and remitted.”Footnote ³

These two inductive rules are to be used in conjunction with two causal rules:

Newton explicitly invokes all four of these rules in deriving the law of universal gravitation in book 3 of the Principia. Very briefly, his derivation starts with six phenomena pertaining to the Keplerian motions of the planets and their satellites that he claims have been astronomically established. Arguing from propositions established in book 1 that show that such motions are produced by an inverse-square attractive force, and then using his causal rules of reasoning 1 and 2, he concludes that this force is the same force in each case that causes the Keplerian motions of all the planets and their moons. In accordance with his rule 3 of inductive generalization, he infers that all bodies in the universe are subject to this force. Finally, by rule 4, since the law is derived from phenomena by induction, it should be considered true, or at least approximately true, until new phenomena are discovered that show that the law needs modification.

Whewell takes on both Newton and Mill in his works. Chapter 13 of Whewell (Reference Woit1847/1967) is devoted to showing what is wrong with each of Newton's four rules and how to substantially reformulate them in Whewellian terms. However, a far greater criticism is reserved for his contemporary Mill in sections of the book that develop his own doctrines of “colligation” and “consilience.” I will discuss Whewell's arguments against Newton and Mill, as well as Norton's arguments against all three.

2. Whewell versus Newton (Mostly)

With respect to Newton's first rule, Whewell's complaint is that it is either trivial and uninteresting or interesting but stifling to science, or else it is quite simply Whewell's own rule of “consilience,” that is, a rule requiring us to find a cause that explains various classes of phenomena, not just the ones we started with. It is trivial, Whewell says, if it is simply construed as telling us to search for true causes—that is, ones that exist (should we search for causes that do not exist?)—and uninteresting because it does not tell how to do this. It is stifling to science if, as Whewell thinks Newton's rule 1 is usually interpreted, it requires us to look for causes only among those “with which we are already familiar.” And it is correct only if construed as conforming to Whewell's own ideas of consilience.

I think Newton would and should have a fit with this. His first rule does not say simply “look for true causes,” although I suppose it is committed to at least that much. It is clear from Newton's formulation, from his brief discussion following that formulation, and from his actual use of that rule in deriving the law of gravity that he has something more in mind, namely, a principle of simplicity: do not infer as true more causes than are needed. Indeed, immediately after formulating the rule, Newton writes: “Nature does nothing in vain, and more causes are in vain when fewer suffice. For nature is simple and does not indulge in the luxury of superfluous causes” (Reference Norton and Achinstein1999, 794). Nor, contrary to Whewell, does Newton restrict inferred causes to ones of a type already known to operate—he simply says that if one suffices, do not postulate two, and that effects of the same kind should be assigned to the same cause—not that these causes must be ones already known.

Whewell has the following objection to Newton's rule 2: it does not tell us much, since it does not tell us how to determine whether effects are “of the same kind.” For example, Whewell asks, why take planetary motions to be motions of the same kind as that of (1) bodies “moving freely [i.e., not propelled by a contact force] in a curvilinear path,” as Newton did, rather than as that of (2) bodies “swept round by a whirling current,” as Descartes did?

Newton has, I would think, two replies. One is a tu quoque. Whewell himself, when he characterizes what he calls the “consilience of inductions,” writes, “But the evidence in favor of our induction is of a much higher and more forcible character when it enables us to explain and determine cases of a kind different from those which were contemplated in the formation of our hypothesis” (Reference Woit1847/1967, 65) Whewell, like Newton, does not tell us how to determine whether cases are of the same kind or a different one. But this is because the question is an empirical one, not one to be settled a priori in advance for all cases. This reply is one with which Whewell should agree. Newton's second response would be to reject Whewell's “loaded” descriptions of the observed effects, such as those in (1) and (2) above—“loaded” in that they presuppose something important about the cause being inferred. In the astronomical case, Newton's same “effects,” from which he infers a common cause, concern observed Keplerian facts about the orbits of the planets and their satellites, and (contra Whewell) not claims such as (1) and (2) about what is or is not producing those orbits (whether the cause is a whirling current or is not a contact force). The effects, as described by Newton, are logically compatible with Newton's central force cause, as well as with Cartesian vortices. Which of these causes to infer is an empirical issue. All Newton is claiming, using his rules 1 and 2, is that the orbital Keplerian “effects” being the same all have the same cause, whatever that is.

Finally, let me mention what Whewell mainly objects to about Newton's rules 3 and 4. According to Whewell, Newton makes it seem that, independently of any other considerations, anytime we find a common property of bodies in the ones examined, we can generalize to all bodies. But, Whewell insists, “the assertion of the universality of any property of bodies must be grounded upon the reason of the case, and not upon any arbitrary maxim” (Reference Woit1847/1967, chap. 13, sec. 14; such as Whewell takes Newton's rule 3 to be). Whewell's point here is an important one. I believe he is saying, more generally, that inductive generalization is not governed by a formal rule that allows you to infer that all As are Bs simply when all the As you have observed are Bs. In Bacon's words, this is “puerile induction.”

Let me discuss this objection by turning to Mill, whom Whewell regards as an even more unenlightened opponent than Newton, at least about scientific methodology. Mill is more explicit about how to respond to Whewell's claim than is Newton, but I think both would respond in the same way.

Recall Mill's definition of induction as “the process by which we conclude that what is true of certain individuals of a class is true of the whole class” (Reference Newton, Cohen and Whitman1872, 188). Mill does not say, and indeed he explicitly denies, that any induction so defined is justified. Whether it is, he says, depends on “the number and nature of the instances” (189). For Mill, and I think for Newton as well, whether a particular inductive inference from “all observed As are Bs” to “all As are Bs” is justified is an empirical issue, not an a priori one. Mill writes that we may need only one observed instance of a chemical fact about a substance to validly generalize to all instances of that substance, whereas many observed instances of black crows are required to generalize about all crows (Mill Reference Newton, Cohen and Whitman1872, 205–6). This is due to the empirical fact that instances of chemical properties of substances tend to be completely uniform, whereas bird coloration, even in the same species, tends not to be. In making an inductive generalization to a law of nature, Mill requires that we vary the instances and circumstances in which they are obtained in a manner described by his “four methods of experimental inquiry.” Whether the instances and circumstances have been varied, and to what extent, is an empirical question, not determined a priori by the fact that all the observed members of the class have some property.

I think it is reasonable to say that Newton, like Mill, is not claiming that any induction from “all observed As are Bs” to “all As are Bs” is valid. For one thing, he never says this. For another, his inductive rule 3 has restrictions, namely, to inductions about bodies and their qualities that “cannot be intended and remitted.” On one interpretation offered by Newton commentators, this rule cannot be applied to qualities such as colors or temperatures of bodies. Moreover, Newton, like Mill, defends his third rule by appeal to a principle of uniformity of nature: “We [should not] depart from the analogy of nature, since nature is always simple and ever consonant with itself” (Reference Norton and Achinstein1999, 795).

Newton does not say how he knows nature is “ever consonant with itself,” or indeed what this entails. Mill is more explicit, though not completely so. Perhaps a little bit charitably, given his discussion in his System of Logic (Reference Newton, Cohen and Whitman1872, 200–206), I would take him to be saying at least, and perhaps not more than, this: there are uniformities in nature, that is, general laws governing various types of phenomena—laws that are inductively and validly inferred, perhaps using Mill's “four methods of experimental inquiry.” That such laws exist is for Mill an empirical claim. Suppose that phenomena governed by such laws bear a similarity to others that by induction we infer are governed by some similar set of laws. Then we can use the fact that one set of phenomena is so governed to strengthen the inference to laws regarding the second set. Whether this is Newton's idea as well, indeed whether it is even all of what Mill has in mind, I cannot say. But I believe at least that neither Newton nor Mill regarded inductive inferences as subject to a purely formal rule allowing one to validly infer that for any sample and population the former matches the latter.

3. Enter Norton

John Norton (Reference Popper2005) argues that there are no valid universal rules of inductive inference at all. He claims that particular inductive inferences are warranted by and only by empirical facts, and not at all by being shown to satisfy some formal schema or template in the manner of formal deductive inferences. I think it is safe to say that he has in his sights Newton's methodological rules and Mill's ideas about induction, among many others. Indeed, I think that Norton would equally reject Whewell's rules pertaining to consilience and coherence, though he would probably agree with a number of Whewell's charges against Newton and Mill. So while Norton is a friend of particular inductions, he is also a foe of any general inductive principles.

Again, I want to take up the challenge to defend Newton and Mill. To begin with, what is the difference between formal rules of the type Norton has in mind in deductive logic and rules of the sort Newton and Mill propose? Are the latter formal in any sense? They can certainly be given what I would regard as a reasonably formal expression. For example, one of Mill's definitions of an inductive inference might be written like this: an inference from a sentence of the form “all observed As are Bs” to one of the form “all As are Bs.” However, as I have already emphasized, what is not the case, and what Mill explicitly denies, is that any argument representing an inference of this form that has true premises will have a conclusion that is true or probably true. Mill makes it clear that there are numerous arguments of this form that are really bad.

Similarly, one might express Newton's causal rule 2 as an inference from “e1 and e2 are effects of the same kind” to “e1 and e2 have the same cause.” However, what is not the case, as Newton recognizes, is that any argument of this form with a true premise will have a conclusion that is true or probably true. Just think of simple cases, such as uniform motion in a straight line, that can be caused by a pair of forces or by the absence of a force.

Are there any analogies in deductive logic? Consider a type of argument form in deduction called the syllogism. One standard way to define a syllogism is as a deductive argument with two premises and a conclusion, each of which has one of the four categorical forms, A, E, I, and O; the argument contains three terms, a different pair of which occurs in each of the three propositions. Put this way, a syllogism may be valid or invalid, as can readily be determined by a Venn diagram. In other words, there are good and bad syllogisms, all having a formal structure satisfying the formal rule defining a syllogism. Of course, you could define a syllogism as a valid argument satisfying the formal definition, in which case there would not be any invalid syllogisms. But then you could do the same for “inductive generalization,” defining it as a valid inference of the form “all observed As are Bs, so all As are Bs.” Then there would be no invalid inductive generalizations. No one, certainly not Mill, wants to do that.

So I will just keep the original definition I gave for the syllogism and point out that, for our purposes, the important difference between this case and the inductive and causal ones is that whether a syllogism is valid can be determined entirely by a priori formal means. Whether an inductive generalization or a causal inference is valid or reasonable cannot be. In view of this, how should we understand inductive rules of the sort proposed by Newton and Mill? One suggestion is to say that while rules of deduction are exceptionless, rules of induction are just “rules of thumb.” They have numerous exceptions, but at the outset at least, when we cannot think of any exceptions at the moment, they are generally to be followed. (This is Paul Feyerabend's one concession to rationality.) If so, we might write Mill's idea of inductive generalization as: if all the As you have observed have been Bs, then, pending exceptions, it is reasonable at the outset at least to assume that all As are Bs. Similarly, we might think of Newton's causal rule 2 as something like this: if effects are of the same kind, then, pending exceptions, it is reasonable at the outset at least to assume they have the same cause.Footnote ⁴ This would make the rules in question pretty weak and rather vague—features of rules that I think both Newton and Mill would prefer to avoid.

I want to suggest a very different way to regard the rules of Newton and Mill. Let me begin with Mill. When Mill (Reference Newton, Cohen and Whitman1872, 188) characterizes induction as “the process by which we conclude that what is true of certain individuals of a class is true of the whole class,” what is he doing? For one thing, as I have said, he is giving a definition of “induction” or “inductive generalization.” And as he himself emphasizes, the definition permits both good and bad inductions. For another, as I have also said, he is claiming that whether the induction is good or bad depends on empirical considerations involving the size and variation of the sample, as well as on other information available. But he is doing something else as well. For Mill, one of the main aims of science is to “discover and prove” causal laws that enable one to explain and predict phenomena. Such discovery and proof requires making inductions to causal generalizations, which is what laws are for him. Accordingly, in giving what Norton calls a “formal inductive template,” what Mill is doing, among other things, is identifying and characterizing an important type of scientific reasoning. He is not justifying any particular instance of reasoning of that form by saying that it has that form—any more than in deductive logic one justifies a particular syllogism by saying that it has the form of a syllogism.

Similarly, when Newton introduces his four methodological rules, he is at least implicitly claiming that causal reasoning of the sort expressed in rules 1 and 2 and inductive reasoning of the sort expressed in rules 3 and 4 are crucial in establishing propositions in empirical science. To be sure, it would have been possible for Newton the scientist to argue for his law of gravity without explicitly invoking any of his four methodological rules. By invoking them, however, one of the things I take Newton to be doing is attempting to give us a better understanding of, as we might put it, the “logic” of his argument. And that, indeed, might help us understand the argument and convince us that it is a good one, that its conclusion does in fact follow from the Newtonian “Phenomena” cited, and how it does, which indeed was one of Newton's primary concerns.

Look, for example, at Newton's initial discussion of proposition 5 of book 3.Footnote ⁵ In this discussion, he claims that the revolutions of the moons of Jupiter about Jupiter, of the moons of Saturn about Saturn, and of Mercury and Venus and the other known planets are phenomena of the same kind as the revolution of our moon about the earth and, therefore (by rule 2), depend on causes of the same kind, especially since it has been proved that the forces on which those revolutions depend are directed toward the centers of Jupiter, Saturn, and the sun and decrease according to the same ratio and law (in receding from Jupiter, Saturn, and the sun) as the force of gravity (in receding from the earth). Rule 2, to which Newton alludes, is his second rule of reasoning according to which “the causes assigned to natural effects of the same kind must be, so far as possible, the same.” Newton is here characterizing what he is doing as arguing from the same kind of effects to the same unique cause. John Norton is right in saying that the empirical force of the argument comes when Newton argues that these are the same type of effects, namely, that these are all Keplerian motions. Newton is saying that in view of this he can infer that they have causes of the same kind, especially since he has already proved in book 1 that such motions must be governed by central inverse-square forces. He is also saying that he is simply arguing from effects of the same kind to causes of the same kind, in accordance with his second rule, which he thinks will help explain what he is doing. If you want to attack Newton the physicist, argue against his empirical idea that these motions are all Keplerian or that such motions are produced by central inverse-square forces. If you want to attack Newton the methodologist, you need to show that he was not making an inference from effects to causes, or that pointing this out in the way that he does is of little, if any, value.

If you think in the latter way, possibly a loose analogy with musicology might help you. The classical sonata has a rather definite formal structure that music theorists teach. It usually has three movements (an allegro, a slow movement, and a closing presto). In the first movement, there is the introductory section, the exposition, the development, the recapitulation, and the coda. And so forth. Simply following this form will not make the resulting sonata good or bad. And, I suppose, it is possible to play a Mozart sonata well without knowing its structure. However, the musicologist helps us understand what is going on in the sonata by explicitly invoking the formal structure and applying it to the particular sonata. So, perhaps, Newton is helping us see the formal structure of his argument by appeal to his rules, in the way the musicologist is helping us understand the Mozart sonata.

4. 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 hypothetico-deductive method, according to which you prove or confirm a hypothesis by deriving observational predictions from it that 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; and 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 (see Achinstein Reference Achinstein1991), or, in our own time, criticisms of string theory by physicists citing methodological grounds similar to those of Newton and Mill (see Smolin Reference Whewell2006; Woit Reference Woit2006).

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 an analogy here between what Mill is doing in the case of inductive inference and what, in the next century, Hempel (Reference Lipton1965) 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—that is, 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, either in 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 is 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—that is, a probability—for every proposition. Newton was a bit crazy, but neither he nor Mill nor John Norton are Bayesians.