The Whewell-Mill Debate in a Nutshell

Malcolm R. Forster and Ann Wolfe, 3/31/98

Note: This is an excerpt from a draft in progress.

What is induction? John Stuart Mill (1874, p. 208) defined induction as the operation of discovering and proving general propositions. William Whewell (in Butts, 1989, p. 266) agrees with Mill’s definition as far as it goes. Is Whewell therefore assenting to the standard concept of induction, which talks of inferring a generalization of the form "All As are Bs" from the premise that "All observed As are Bs"? Does Whewell agree, to use Mill’s example, that inferring "All humans are mortal" from the premise that "John, Peter and Paul, etc., are mortal" is an example of induction? The surprising answer is "no". How can this be?

Our reading of Whewell is that he has an entirely different understanding of the term "general". For Whewell, the proposition "All As are Bs" is not general if it is a mere juxta-position of particular cases (see Butts, 1989, p. 163). Rather, for Whewell it is necessary that (Butts, 1989, p. 47) "In each inductive process, there is some general idea introduced, which is given, not by the phenomena, but by the mind." The proposition is constituted of facts and conceptions which are, "bound together so as to give rise to those general propositions of which science consists" (ref--see MR). For Whewell "All humans are mortal" is not general in the appropriate sense because there has been no conception added. Whewell insists that in every genuine induction, "The facts are known but they are insulated and unconnected . . . The pearls are there but they will not hang together until some one provides the string" (Butts 1989, pp. 140-141). The "pearls" are the data points and the "string" is a new conception that connects and unifies the data. The "pearls" in "All As are Bs" are unstrung because "All As are Bs", though a general proposition in the sense that it applies to all relevant instances, does not connect or unify the facts; or as Whewell puts it, it does not colligate the facts. Therefore, the standard view of induction is not Whewell’s view of induction.

Whewell distinguishes "colligations" from what are commonly thought of as inductions to make the point that it is only the former that, by virtue of the mental act of introducing conceptions, genuinely connects or unifies facts. To make this point, he gives an example that illustrates the subtle ways in which our perceptions interact with our mental conceptions, an example that appeals to our intuitions even in simple cases where that interaction might not be so obvious at first glance.

When anyone has seen an oak-tree blown down by a strong gust of wind, he does not think of the occurrence any otherwise than as a Fact of which he is assured by his senses. Yet by what sense does he perceive the Force which he thus supposes the wind to exert? By what sense does he distinguish the Oak-tree from all other trees? It is clear upon reflexion, that in such a case, his own mind supplies the conception of extraneous impulse and pressure, by which he thus interprets the motions observed, and the distinction of different kinds of trees. . . . The Idea of Force, and the idea of definite Resemblances and Differences, are thus combined with the impressions on our senses and form an indistinguished portion of that which we consider as the Fact. (Butts, pp.123-124)

So while Mill and Whewell agree that inductions produce propositions that are more general than the premises from which they are generated, both the meaning of "generality" and the nature of the inductive process remain important points of contention between them.

Does this dispute over the nature of induction make a difference in scientific examples? Whewell and Mill argued over Kepler’s discovery of the elliptical motion of Mars. Kepler started with observations of the position of Mars relative to the sun at various times. These observations might be represented as points scattered around the sun, from which Kepler inferred that Mars’s orbit is an ellipse. As in any example of curve-fitting, the data points are the pearls and the curve is the string.

Could this example be understood as an example of induction in Mill’s sense? Whewell and Mill can agree that the conclusion of the inference is that "All positions of Mars lie on ellipse b", where b is the name of a particular ellipse. So, in this example, the predicate A is "is a position of Mars" and B is "lies on ellipse b." But Mill has to say that the data are of the form "at time t1 Mars lies on ellipse b, at time t2 Mars lies on ellipse b, and so on." Notice that for Mill, the predicates that appear in the general proposition also appear in the description of the data.

On the other hand, Whewell considers the data to contain no mention of the ellipse b, or any ellipse, so "lying on ellipse b" is a new conception that colligates the data. The data are "at time t1 Mars is at position x1, at time t2 Mars is at position x1, and so on." For Whewell, the facts and the conception are then "bound together so as to give rise to those general propositions of which science consists" (ref--see MR). So, Kepler’s conclusion is general in the sense that the general conception of an ‘ellipse’ is "superinduced" upon the facts, and is not a "mere union of parts" or a "mere collection of particulars." (Butts, 1989, p. 163.) That is why Whewell sees Kepler’s inference as a colligation and therefore, a genuine induction.

Against Mill, Whewell (1849, p.//) says that "the fact of the elliptical motion was not merely the sum of the different observations, is plain from this, that other persons, and Kepler himself before this discovery, did not find it by adding together the observations." When Millians see this quote, they interpret it wrongly as saying that an inductively inferred proposition is not merely the sum of the observations because it covers new instances. But Whewell does not agree that the observations contain, or determine, the conception of an ellipse. So, Mill and Whewell have substantively different views of Kepler’s inference even if they both agree that it is an induction.

We have already seen that the mortality example is an induction for Mill, but not for Whewell. But are there cases that Whewell would see as inductions but Mill would not? Such examples are possible, for Mill insists that "any process in which what seems the conclusion is no wider than the premises from which it was drawn, does not fall within the meaning of the term" (p.210). So imagine that an economist wants to explain the inflation rate of the Soviet Union, and uses the known inflation rates for all the years from 1917 to 1990. For Whewell, this can count as a genuine inductive inference if the explanation introduces a new conception; maybe the concept of price control. But Mill would be forced to say that there is no induction because there are no new instances of the proposition, due to the unfortunate disintegration of Soviet Union 1990. We think that intuition is on Whewell’s side in this debate, for the induced proposition does have implications about what the inflation rate would have been in 1991 had the Soviet Union survived. But for Mill this is not enough because this is not a testable prediction.

To sum up: Whewell introduced the term "colligation" to refer to the process of conceptualizing observational data. This is the essential part of induction for Whewell and he used the terms "induction" and the "colligation of facts" interchangeably. Mill agreed with most of what Whewell had to say about colligation, but viewed this as a process that occurs separate from and prior to genuine induction.

This issue is especially salient at a time when various versions of and extensions of hypothetico-deductivism, such as Popper’s falsificationism, logical positivism, and even Bayesianism, have come under increasingly severe criticism. Whewell’s philosophy of science presents an alternative to the modern trend, lead in part by Kuhn (1970). However, Whewell’s alternative is especially interesting, and attractive, because it does not replace the philosophy of science with, or reduce it to, a sociology of scientific communities.

Is this more than a terminological dispute? Is it substantive? When added to claims about justification, the dispute is clearly substantive. Whewell claims that the colligation is an essential to the consilience of inductions, which is essential to the justification of scientific theories. Mill disagrees. The main thesis of this paper is that Whewell is right and Mill is wrong.

Colligation, for Mill, is a part of the discovery process, or the process of invention, whereas induction is relevant to questions of justification. Whewell’s characterization of induction, Mill objects, belongs to (what we might call) the ‘context of discovery,’ and Mill thinks that Whewell confuses them. Accordingly Mill (1874, p. 222) charges that "Dr Whewell calls nothing induction where there is not a new mental conception introduced and everything induction where there is." "But," he continues, "this is to confuse two very different things, Invention and Proof." "The introduction of a new conception belongs to Invention: and invention may be required in any operation, but it is the essence of none." Abstracting a general proposition from known facts without concluding anything about unknown instances, Mill goes on to say, is merely a "colligation of facts" and bears no resemblance to induction at all. Whewell, of course, disagrees.

True, Whewell does think that mental acts are essential features at every stage of scientific progress, and that mental acts are essential to invention or discovery. But to say that they are essential to discovery does not imply that they are not also essential to justification. So, Mill has no good reason to accuse Whewell of confusing invention and proof. In fact, Whewell concerns himself extensively with delineating between invention (i.e. colligations) and justification (i.e., Consiliences of Inductions). As we shall see below, Whewell’s notion of consilience requires that the conceptions involved in an induction have to agree, or jump together, in order for a consilience to occur. For Whewell, colligation is the essential feature of any induction, and is therefore essential to any consilience of inductions, and therefore essential to the justification of theories since consiliences constitute such justification. This is why the context of discovery is an inseparable part of the context of justification for Whewell. But Whewell never confuses discovery and justification; he is clear in his view that a colligation is not justification by itself.

To bolster our claim, we emphasize that Whewell certainly does not think that all inductions are justified. Whewell acknowledges that in the course of science, "Real discoveries are . . . mixed with baseless assumptions" (Butts 1989, p. 145). That is precisely why Whewell considers a consilience of inductions necessary to provide convincing evidence for a hypothesis’ validity. The validity of a hypothesis, for Whewell, remains suspect even if it appears to accurately predict new instances of the same kind (as in the mortality example and the Kepler example). That is unless, or until, it passes his most rigorous test by leading to a consilience. The inductive inference that Mill seems to accept as justification -- that a proposition hold when tested against cases of the same kind -- is only one of several tests in Whewell’s methodological schema. In fact, Whewell, by requiring more, and increasingly more difficult, tests insists upon far more justification for a theory than does Mill.

In place of the consilience of inductions, Mill talks about the deductive subsumption of lower level empirical laws under more fundamental laws, which is a well-known part of hypothetico-deductivism. Whewell’s account of consilience gets around the common objection that deductive subsumption is too easy to satisfy. For instance, hypothetico-deductivism tries to maintain that Galileo’s theory of terrestrial motion, call it G, and Kepler’s theory of celestial motion, K, are subsumed under Newton’s theory N because N deductively entails G and K. A common objection is that the alleged deductive relations hold only ‘approximately’, but there is a more serious problem for this view. The problem is that G and K are also subsumed under the mere conjunction of G & K, so subsumption by itself does not capture the idea that N is more unified or consilient. Fortunately, Whewell’s view does not have that consequence because consilience makes essential reference to the fact that Newton successfully colligated facts about motion using the added conceptions of force and mass.

There is a secondary dispute between Mill and Whewell, which is seen as a red herring in light of this analysis: It doesn’t matter whether conceptualization occurs prior to an induction as Mill insists, or during the induction as Whewell maintains. No matter which view is correct, the important difference is that conceptualization plays an essential justificatory role for Whewell, but not for Mill.

However, there is another closely related issue that is relevant. Mill claims that the property of "lying on ellipse b" is determined by and read from the data themselves. If this empiricist view of concepts is correct, then it is hard to see why conceptualization should matter to theory comparison, because all competing theories will be on the same footing. According to Mill (1874, p.216, Mill’s emphasis):

Kepler did not put what he had conceived into the facts, but saw it in them . . . A conception implies, and corresponds to, something conceived: and though the conception itself is not in the facts, but in our very mind, yet if it’s to convey any knowledge relating to them, it must be a conception of something which really is in the facts . . ."

Whewell does not deny that the regularities in nature exist before we perceive or conceive them; he does not reject the claim that the orbit of Mars was elliptical before anyone knew that to be true. Rather, Whewell thinks that Kepler placed the data "in a new system of relations with one another" was not determined by the data themselves. The interpretation of the data is theory-dependent. This is especially clear in curve-fitting examples, which Whewell talks about in some detail (see Butts, 1979, pp. 211-237). According to Whewell, "the Colligation of ascertained Facts into general Propositions" consists of (1) the Selection of the Idea, (2) the Construction of the Conception, and (3) the Determination of the Magnitudes. In curve fitting, these three steps correspond to (1) the determination of the Independent Variable, (2) the Formula, and (3) the Coefficients. In the Kepler example the independent variable is ‘time’. The data are observations of Mars at various times, and the aim of the induction is to characterize all positions as a function of time. The second step introduces the conception of an ellipse. At that stage, the claim is that the orbit of Mars is some ellipse, without saying which ellipse. In the third step, the family of ellipses is fitted to the data, and the measured parameters (or coefficients as Whewell calls them) are those characterizing the best fitting ellipse. This is ellipse b, and this third step yields the specific claim that all points on Mars’ orbit lie on ellipse b. There are two important points to notice. First, the data first enter the process in step 3, but this process makes no sense unless the formula is already fixed because the "best fitting curve" means "the best fitting curve in a family." If a different formula were chosen, then the resulting orbit would not be an ellipse. Moreover, it is always possible that a different family, or formula, could yield a curve that fits equally well. So, there is no sense in which the data determines the formula. Mill’s idea of the data is that "all observed positions of Mars lie on ellipse b" has no logically or historically basis.

What Mill misses is the important distinction that Whewell makes between the idea(s) that are used to express the facts and the conception(s) used to colligate the facts. As we have seen, on Whewell’s view, all facts are mind-laden to the core. But the data are laden with ideas, which are a special class of conceptions, like time, space, and number. These are needed in step 1 to determine the independent variables. It may be that, for Whewell, ideas are determined by the data, but they are not to be confused with the new conception that must be added in step 2. There is no sense in which the conception used to colligate facts is determined by the data.

Mill’s view is especially implausible in other examples like Newton’s argument for universal gravitation (see Forster 1988, section 5). How can the conceptualization occur prior to the induction properly so-called in such an example? Mill would have to say that Newton saw the masses and forces in the data prior to the induction. In other words, Mill would have to assume that Newton observes that the planets are at the positions they would have if Newton’s theory of gravitation were true. But that is a consequence of the inferred proposition, and not part of the description of the data from which the inference began. We do not need to labor this point because it is now widely accepted that theories are underdetermined by the facts.

To represent Whewell as interested only in the psychology of discovery or to characterize 19th century empiricists like Mill as denying to the mind any role in the development of scientific knowledge, is to oversimplify the deep philosophical differences between Whewell and his critics. To do so also downplays Whewell’s innovative, albeit controversial, contributions to the problem of justification in science (see Section IV).We have seen that there are fundamental and important differences between Mill’s and Whewell’s philosophy of science and that the nature and the substance of those differences are not merely terminological or as obvious as they may seem at first glance. But contemporary writers have also misunderstood fundamental aspects of Whewell’s epistemology and been led to a confused reading of his philosophy of science more generally. In the following sections we will attempt to clear up some of those misunderstandings. …

 

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