WHEWELL VERSUS MILL: ARE INDUCTIVE PREMISES MERE DESCRIPTIONS OF THE DATA?

A second, related claim that Whewell makes against Mill is that inductive premises are not mere descriptions or summaries of the data. According to Whewell, Kepler's ascription of an “elliptical orbit” to Mars is an induc­tion.

I suggest that the disputants here are referring to different things. Mill is counting the following claim as a description and not an inductive generalization:

M: All observed points of Mars lie on an ellipse.

11. Mill (1872, p. 186) writes that:

the process of indirectly ascertaining individual facts is as truly inductive as that by which we establish general truths. But it is not a different kind of induction; it is a form of the very same process: since... whenever the evidence which we derive from observation of known cases justifies us in drawing an inference respecting even one unknown case, we should on the same evidence be justified in drawing a similar inference with respect to a whole class of cases.

Mill calls this a summary of separate observations. More generally, if we note that all the observed data points in a given experiment lie on a given curve, Mill is saying, we are simply summarizing the observed data by “obtaining a representation of the phenomena as a whole, by combining, or as we may say, piecing these detached fragments together.” For Mill this is not an inductive conclusion because the claim is not that all the points—both observed and unobserved—lie on the curve. Indeed, as far as the descriptive or summarizing claim is concerned, there are many dif­ferent representations of the observed data points; many different curves are satisfied by these points.^[43]

What Whewell is claiming is an inductive conclusion is not M as shown previously but

W: The orbit of Mars is an ellipse.

Whewell makes it clear that by the “orbit of Mars” he means the orbit consisting of both observed and unobserved points in the orbit.^[44] Accord­ingly, Whewell is not claiming that Mill's (M) is an inductive conclusion, nor is Mill denying that Whewell's (W) is an inductive conclusion (since it is a generalization from observed points to all points). Mill wants to call (M) a description of observed facts, whereas Whewell rejects this because it does not recognize that Mars moves in an orbit.

There are several points of disagreement here. One, mentioned earlier, is that Mill's talk of “describing” and “summarizing” the observed data makes it sound as if the description or summary can always be directly read off the data without reasoning or calculation. Whewell, by contrast, rejects this idea: the description of the observed positions of Mars as lying on an ellipse is “given, not by the phenomena, but by the mind.” Mill (1872, p. 199) responds by agreeing with Whewell's claim that Kepler's statement that the observed positions lie on an ellipse “was not the sum of the observations merely; it was the sum of the observations seen under a new point of view. But it was not the sum of more than the observations.”

Whatever Mill intended here, he need not have committed himself to the view that a “description” or “summary” of observed data requires some direct “reading off” of the description from the data—without any reasoning or calculation.^[45] Suppose I record some data points from an experiment to determine how a physical quantity y changes with another physical quantity x. I make three observations. One way to describe or summarize the observed data is to give three pairs of numbers, in which the first represents an observed value for quantity x and the second a cor­responding observed value for y. For example:

Description 1: The three observed points are (1,7), (2,10), (3,13).

Another way to describe or summarize the observed data is as follows:

Description 2: The three observed points in which the x-values are 1, 2, and 3 all lie on the linear curve y = 3x + 4.

Description 2 requires calculation and reasoning in a way that description 1 does not, or at least it requires more such reasoning and calculation. Description 2 might well be spoken of as summarizing the observations reported in description 1 “under a new point of view”—one that requires the mind to go beyond what is “directly observable.” Mill should be willing to concede this point to Whewell.^[46]

Even with such a concession, however, there remains a significant difference in the approach of Mill and Whewell toward induction as a process of reasoning. Mill wants to claim that (M) represents a crucial step in the inductive process—one in which Kepler represents the ob­served positions of Mars using a concept (such as that of an elliptical orbit) in a way that does not make a generalized claim: it does not make a claim about the unobserved positions of Mars or about Mars's continuing to take an elliptical path.^[47] Whewell, by contrast, seems to be denying this step in the induction. Given the observed positions of Mars, he appears to be saying that what Kepler did was apply the concept ellipse to the orbit of Mars, i.e., to the unobserved as well as the observed positions. In other words, Mill claims that Kepler's reasoning involves these steps:

(1) Here are the observed (relative) positions of Mars.

(2) All of these positions lie on an ellipse.

(3) Therefore, all the positions of Mars lie on an ellipse; i.e., the orbit is elliptical.

For Mill, step 2 is not an inductive generalization since it makes no com­mitment concerning the unobserved points. It is simply one (of possibly many) representation of the data in (1). By contrast, Whewell seeks to omit step (2), going directly from (1) to (3), since, on his view, the mind supplies the concept “elliptical orbit,” which it imposes on all the points whether observed or unobserved. Mill is saying that it is possible to repre­sent the observations by using the concept ellipse, as he does in step (2), without making an inductive generalization.

Moreover, representing the observed positions noted in (1) as Mill does in (2) allows Mill to characterize inductive reasoning as “a process by which we conclude that what is true of certain individuals in a class is true of the whole class.” Going directly from step (1) to step (3), as Whewell does, allows him to characterize induction as a process of going from facts about particulars, whether or not these facts are represented by propositions of the form “all observed As are Bs,” to a general proposition. Whewell (1967, pp. 239-240) says that he agrees with Mill when the latter characterizes induction as “the operation of discovering and forming general proposi­tions” and also as “generalization from experience.”What Whewell wants to do, however, is include among inductions inferences from particulars, how­ever the latter are characterized, to a general proposition. Mill will insist that such inferences can be called inductive only when the information about the particulars is, or can be, put in the form of a “description,” such as (2).

Is this simply a difference in terminology—Mill using the term “induc­tion” only for cases that are, or can be, put into the form he wants and Whewell using the term more broadly? It is a difference in terminology, but one that, I think, reflects a difference in viewpoint about induction. Unless one is able to express an inductive inference to a generalization so as to include a premise such as (2), Mill is saying, we may be unable to see the connection between a premise such as (1) and the conclusion (3). Why should these observed relative positions of Mars justify the claim that the Martian orbit is elliptical? The answer is supplied by premise (2).

Even more important, Mill would say, characterizing induction the way Whewell has done, and even imposing Whewell's crucial “consilience” condition for defending the conclusion of an induction, will not suffice to produce an argument whose premises justify its conclusion.

(A) Mars has been observed to exhibit phases, like the moon.

Therefore,

The Martian orbit is elliptical.

(B) Bodies have been observed to fall toward the earth with approximately uniform acceleration.

Therefore,

All bodies in the universe attract each other in accordance with an inverse-square force.

In both cases the premise concerns “particulars” that have been observed. The premise and conclusion are true. And Whewell's requirements for a good induction appear to be satisfied (the conclusion can be used to ex­plain and predict a range of facts in addition to the ones reported in the premise). The problem, Mill would say, is that in each case the premise does not warrant the conclusion. What is missing in both is a suitable de­scription of numerous instances of items satisfying the general conclusion (e.g., in (B) instances of bodies attracting each other with an inverse­square force).

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