EXPLANATORY ANOMALIES

One objection critics of my account may raise is that it does not do justice to explanatory anomalies in the wave theory. That theory was not able to explain all known optical phenomena.

Let me now say how wave theorists could in principle deal with such anomalies that relate to the probabilistic reconstruction I offer. The sug­gestion I will make is, I think, implicit in their writings, if not explicit. And interestingly, it is a response that combines certain Whewellian and Millian ideas. In what follows, I restrict the anomalies to phenomena which have not yet been derived from the wave theory by itself or from that theory together with auxiliary assumptions whose probability is very much greater given the wave theory than without it.

As Cantor notes in his very informative book Optics after Newton:

Probably the central, and certainly the most repeated, claim [by the 1830's] was that in comparison with its rival the wave theory was more successful in explaining optical phenomena. (Cantor 1983, p. 192)

Cantor goes on to cite a table constructed in 1833 by Baden Powell, a wave theorist, listing 23 optical phenomena and evaluating the explana­tions proposed by wave and particle theories as “perfect,” “imperfect,” or “none.” In the no-explanation category there are 12 entries for the particle theory and only 2 for the wave theory; while there are 18 “perfects” for the wave theory and only 5 for the particle theory.

Appealing, then, to the explanatory success of the wave theory, a very simple argument is this:

Optical phenomena O_j,...,O_n can be coherently explained by the wave theory.

O is another optical phenomenon. (6)

So probably

O can be coherently explained by the wave theory.

By a “coherent” explanation I follow what I take to be Whewell's idea: either the phenomenon is explained from the theory without introducing any additional assumptions, or if they are introduced they cohere both with the theory and with other known phenomena. In particular, no auxiliary assumption is introduced whose probability given the theory is very high but whose probability on the phenomena alone is low. Or, more generally, no such assumption is employed whose probability on the theory is very much greater than its probability without it.

Commenting on argument (6), the particle theorist might offer a similar argument to the conclusion that the particle theory can also explain O. But this does not vitiate the previous argument. For one thing, by the 1830s, even though Powell's table was not constructed by a neutral observer, it was generally agreed that the number of optical phenomena known to be co­herently explainable by the wave theory was considerably greater than the number explainable by the particle theory. So the wave theorist's argument for his conclusion would be stronger than the particle theorist's for his. But even more importantly, the conclusion of the argument is only that O can be coherently explained by the wave theory, not that it cannot be coher­ently explained by the particle theory. This is not eliminative reasoning.

Argument (6) might be construed in Millian terms as inductive: con­cluding “that what is true of certain individuals of a class is true of the whole class,” and hence of any other particular individual in that class (Mill 1959, p. 188; see note 3 here). Mill's definition is quite general and seems to permit an inference from the explanatory success of a theory to its continued explanatory success.

Argument (6) might also be construed as exhibiting certain Whewel- lian features. Whewell stresses the idea that a theory is a historical entity which changes over time and can “tend to simplicity and harmony.” One of the important aspects of this tendency is that “the elements which we require for explaining a new class of facts are already contained in our system.” He explicitly cites the wave theory, by contrast to the particle theory, as exhibiting this tendency. Accordingly, it seems reasonable to suppose that it will be able to coherently explain some hitherto unex­plained optical phenomenon. The important difference between Whewell and Mill in this connection is not over whether the previous explanatory argument (6) is valid, but over whether from the continued explanatory success of the wave theory one can infer its truth. For Whewell one can; for Mill one cannot.

Let me assume, then, that some such argument as (6) was at least im­plicit in the wave theorists' thinking and that it would have been endorsed by both Mill and Whewell. How, if at all, can it be used to supplement the probabilistic reconstruction of the wave theorists' argument that I offer earlier in the chapter? More specifically, how does it relate to the question of determining the probability of the wave theory given all the known optical phenomena, not just some subset?

The conclusion of the explanatory success argument (6) is that the wave theory coherently explains optical phenomenon O.

Accordingly, we have:

p(W coherently explains optical phenomenon O/

(7)

W coherently explains optical phenomena O₁,...,O_n) > k

where k is some threshold of high probability, and W is the wave theory. If we construe such explanations as deductive, then

“W coherently explains O” entails that p(W/O&O_J,..., O ) > p(W/Oj,..., On) ” (8)

So from (7) and (8) we get the second-order probability statement

p(p(W/O&O₁,...,O_n) > p(W/O₁,...,O_n)/W coherently

explains O₁,...,O_n) > k

But the conclusion of the wave theorists' argument is

p(W/Oj,...,On) « 1, (10)

where O_j,...,O includes all those phenomena for which the wave the­orist supplies a coherent explanation (I suppress reference to background information here). If we add (10) to the conditional side of (9), then from (9) we get

p(p(W/O&O₁,...,O_n) ~ 1/W coherently explains O_p...,O_n and p(W/O_n...,O_n) ~ 1) > k. (11)

This says that, given that the wave theory coherently explains optical phenomena O₁,...,O_n, and that the probability of the wave theory is close to 1 on these phenomena, the probability is high that the wave theory's probability is close to 1 given O—the hitherto unexplained optical phe­nomenon—together with the other explained phenomena. If we put all the known but hitherto unexplained optical phenomena into O, then we can conclude that the probability is high that the wave theory's proba­bility is close to 1 given all the known optical phenomena.

How is this to be understood? Suppose we construe the probabilities here as representing degrees of reasonableness of belief (see addendum to chapter 1). Then the first-order probability can be understood as represent­ing how reasonable it is to believe W; while the second-order probability is interpreted as representing how reasonable it is to believe it is so rea­sonable. This, of course, does not permit the wave theorist to conclude that p(W/O&O₁,...,O_n) ~ 1, i.e., that the probability of the wave theory on all known optical phenomena—explained and unexplained—is close to 1. But it does permit him to say something stronger than simply that his theory is probable given a partial set of known optical phenomena. It goes beyond a wait-and-see attitude with respect to the unexplained phenomena.