THE WAVE THEORISTS' ARGUMENT

Nineteenth-century wave theorists frequently employed the following strategy in defense of their theory.

1. Start with the assumption that light consists either in a wave motion transmitted through a rare, elastic medium pervading the universe, or in a stream of particles emanating from luminous bodies.

2. Show how each theory explains various optical phenomena, including the rectilinear propagation of light, reflection, refraction, diffraction, Newton's rings, polarization, and so on.

3. Argue that in explaining one or more of these phenomena the particle theory introduces improbable auxiliary hypotheses but the wave theory does not. For example, light is diffracted by small obstacles and forms bands both inside and outside the shadow. To explain diffraction particle theorists postulate both attractive and repulsive forces emanating from the obstacle and acting at

a distance on the particles of light so as to turn some of them away from the shadow and others into it. Wave theories such as Young and Fresnel argue that the existence of such forces is very improbable. By contrast, diffraction is explainable from the wave theory (on the basis of Huygens's principle that each point in a wave front can be considered a source of waves), without the introduction of any new improbable assumptions. Similar arguments are given for several other optical phenomena, including interference and the constant velocity of light.

4. Conclude from steps 1 through 3 that the wave theory is true, or very probably true.

This represents, albeit sketchily, the overall structure of the argument.

To be sure, neither Whewell's methodology nor Mill's precludes com­parative judgments. For example, Whewell explicitly claims that the wave theory is more consilient and coherent than the particle theory. And Mill (who believed that neither theory satisfied his crucial inductive step) could in principle allow the possibility that new phenomena could be discovered, permitting an induction to one theory but not the other. I simply want to stress at the outset that the argument strategy of the wave theorists, as I have outlined it so far, is essentially comparative. The aim is to show at least that the wave theory is better, or more probable, than the rival particle theory.

Is the wave theorist's argument intended to be stronger than that? I believe that it is. Thomas Young, both in his 1802 and 1803 Bakerian lectures (reprinted in Crew 1900), makes it clear that he is attempting to show that hitherto performed experiments, and analogies with sound, and passages in Newton, provide strong support for the wave theory, not merely that the wave theory is better supported than its rival. A similar at­titude is taken by Fresnel, whose aim is not simply to show that the wave theory is better in certain respects than the particle theory, but that it is acceptable because it can explain various phenomena, including diffrac­tion, without introducing improbable assumptions; by contrast, the par­ticle theory is not acceptable, since it cannot. Even review articles are not simply comparative. Although he does compare the merits of the wave and particle theories in his 1834 report, Humphrey Lloyd makes it clear that this comparison leads him to assert the truth of the wave theory.

there is thus established that connexion and harmony in its parts which is the never failing attribute of truth.... It may be confidently said that it pos­sesses characters which no false theory ever possessed before. (1877, p. 79)^[50]

Let us now look more closely at the three steps of the argument leading to the conclusion. Wave theorists who make the assumption that light consists either of waves or particles do not do so simply in order to see what follows. They offer reasons, which are generally of two sorts. First, there is an argument from authority: “Leading physicists support one or the other assumption.” Second, there is an argument from some observed property of light. For example, Lloyd notes that light travels in space from one point to another with a finite velocity, and that in nature one observes motion from one point to another occurring by the motion of a body or by vibrations of a medium.

Whatever one might think of the validity of these arguments, I sug­gest that they were being offered in support of the assumption that light consists either of waves or of particles. This is not a mere supposition. Argument from authority was no stranger to optical theorists of this pe­riod. Young in his 1802 paper explicitly appeals to passages in Newton in defense of three of his four basic assumptions. And Brougham, a particle theorist, defends his theory in part also by appeal to the authority and success of Newton. Moreover, the second argument, if not the first, can reasonably be interpreted as an induction in Mill's sense, i.e., as claiming that all observed cases of finite motion are due to particles or waves, so in all probability this one is too.^[51]

I suggest, then, that wave theorists offered grounds for supposing it to be very probable that light consists either of waves or particles. I will write their claim as

p(B/A&C) = p1 (1)

where W is the wave theory, P is the particle theory, O includes certain observed facts about light including its finite motion, and b is background information including facts about modes of travel in other cases.

This is the first step in the earlier argument. I will postpone discus­sion of the second step for a moment, and turn to the third. Here the wave theorists assert that in order to explain various optical phenomena the rival particle theorists introduce improbable auxiliary hypotheses. By contrast, the wave theorists can explain these phenomena without intro­ducing auxiliary hypotheses, or at least any that are improbable. Why are the particle theorists' auxiliary hypotheses improbable? And even if they are, how does this cast doubt on the central assumptions of the particle theory?

Let us return to diffraction, which particle theorists explained by the auxiliary hypothesis that attractive and repulsive forces emanate from the diffracting obstacle and act at a distance on the light particles, bending some into the shadow and others away from it. By experiment Fresnel showed that the observed diffraction patterns do not vary with the mass or shape of the diffracting body. But known attractive and repulsive forces exerted by bodies do vary with the mass and shape of the body. So Fresnel concludes that the existence of such forces of diffraction is highly im­probable. Again it seems plausible to construe this argument as an induc­tive one, making an inference from properties of known forces to what should be (but is not) a property of the newly postulated ones. Fresnel's experiments together with observations of other known forces provide inductive reasons for concluding that the particle theorists' auxiliary as­sumption about attractive and repulsive forces is highly improbable.

Even if this is so, how would it show that other assumptions of the particle theory are improbable? It would if the probability of the auxiliary force assumption given the other assumptions of the particle theory is much, much greater than the probability of this auxiliary assumption not given the rest of the particle theory, i.e., if

p(B IA&C₁) = 1 (2)

where A is the auxiliary assumption, O includes information about dif­fraction patterns and Fresnel's experimental result that these do not vary with the mass or shape of the diffractor, b includes information about other known forces, and means “is much, much greater than.” If this condition is satisfied, it is provable that the other assumptions of the par­ticle theory have a probability close to zero,^[52] i.e.,

p(BM&C₃) = 1 (3)

Although wave theorists did not explicitly argue for (2) above, they clearly had grounds for doing so. If by the particle theory P light consists of particles subject to Newton's laws, and if by observational results O light is diffracted from its rectilinear path, then by Newton's first law a force or set of forces must be acting on the light particles.

From (1) and (3) we infer:

P (W/O&u)»1, (4)

that is, the probability of the wave theory is close to 1, given the back­ground information and certain optical phenomena, including diffraction.

Now we can return to the second step of the original argument, the one in which the wave theorist shows that his theory can explain a range of optical phenomena, not just the finite velocity of light and diffraction. What inferential value does this have? The wave theorist wants to show that his theory is probable not just given some limited selection of optical phenomena but given all known optical phenomena. This he can do if he can explain these phenomena by deriving them from his theory. Where Op...,O represent known optical phenomena other than diffraction and the constant velocity of light—including rectilinear propagation, reflection, refraction, and interference—if the wave theorist can derive these from his theory, then the probability of that theory will be at least sustained if not increased. This is a simple fact about probabilities.

Accordingly, the explanatory step in which the wave theorist derives various optical phenomena O1,...,O from his theory permits an infer­ence from (4) above to:

p( W/O 1,...,O_n&O&b)« 1, (5)

i.e., the high probability of the wave theory given a wide range of ob­served optical phenomena.

If the explanation of known optical phenomena sustains the high prob­ability of the wave theory without increasing it, does this mean that such phenomena fail to constitute evidence for the wave theory? Not at all. Ac­cording to a theory of evidence I have developed (see chapter 1), optical phenomena can count as evidence for the wave theory even if they do not increase its probability. I reject the usual increase-in-probability account of evidence in favor of conditions that require the high probability of the theory T given the putative evidence O_i, and the high probability of an explanatory connection between T and O., given T and O.. Both condi­tions are satisfied in the case of the wave theory.

In formulating the steps of the argument in the probabilistic manner above, I have clearly gone beyond what wave theorists say. For one thing, they do not appeal to probability in the way I have done. More impor­tantly perhaps, while they argue that auxiliary hypotheses of the particle theorists are very improbable, they do not say that these assumptions are much more probable given the rest of the particle theory than without it. The following points are, I think, reasonably clear, (i) Wave theorists suppose that it is very likely that the wave or the particle theory is true, an assumption for which they have arguments. (ii) They argue against the particle theory by criticizing auxiliary assumptions of that theory, which in­troduce forces (or whatever) that violate inductively supported principles. (iii) Wave theorists argue that their theory can explain various optical phenomena without introducing any such questionable assumptions. (iv) Their reasoning, although eliminative, is different from typical elimina­tive reasoning; their first step is not to canvass all possible theories of light, but only two, for which they give arguments; their reasoning is not of the typical eliminative form “these are the only possible explanations of optical phenomena, all of which but one lead to difficulties.” Reconstruct­ing the wave theorists' argument in the probabilistic way I have done captures these four points. Whether it introduces too many fanciful ideas is a question I leave for my critics.

Is the argument Whewellian or Millian? It does satisfy the first three of Whewell's conditions. It invokes the fact that various optical phenomena are derived from the wave theory. These include ones that prompted the theory in the first place (rectilinear propagation, reflection, and re­fraction), hitherto unobserved phenomena that were predicted (e.g., the Poisson spot in diffraction), and phenomena of a kind different from those that prompted it (e.g., diffraction, interference, polarization). The argument does not, however, satisfy Whewell's fourth condition. It does not appeal to the historical tendency of the theory over time to become simpler and more coherent. But the latter is not what divides Whewell from Mill. Nor is it Whewell's first three conditions, each of which Mill allows for in the ratiocinative part of his deductive method. Mill's claim is only that Whewell's conditions are not sufficient to establish the truth or high probability of an hypothesis. They omit the crucial first step, the inductive one to the hypothesized causes and laws.

As I have reconstructed the wave theorists' argument, an appeal to the explanatory power of the theory is a part, but not the whole, of the reasoning. There is also reasoning of a type that Mill would call inductive. It enters at two points. It is used to argue that light is most probably com­posed either of waves or of particles (e.g., the “finite motion” argument of Lloyd). And it is used to show that light is probably not composed of particles, since auxiliary hypotheses introduced to explain various optical phenomena are very improbable. This improbability is established by in­ductive generalization (e.g., in the case of diffraction, by inductively gen­eralizing from what observations and experiments show about diffraction effects, and from what they show about forces). My claims are that wave theorists did in fact employ such inductive reasoning; that with it the argument that I have constructed is valid; and that without it the argu­ment is invalid, or at least an appeal to Whewell's explanatory conditions is not sufficient to establish the high probability of the theory (though this last claim requires much more than I say here; see Achinstein 1991, Essay 4).

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