IS THE ARGUMENT FOR MOLECULES CIRCULAR?

The basic structure of Perrin's reasoning seems to be this:

1. From various assumptions, including that molecules exist, and that gases containing them satisfy the ideal gas law, Perrin derives equation (7), which relates the number of molecules at a height h in a container of gas to Avogadro's number and to other quantities including the mass of a gas molecule and the temperature of the gas.

2. Perrin then claims that this formula, or a variation of it, can be applied to visible Brownian particles suspended in a fluid, yielding equation (9).

3. Next he devises ways to experimentally measure the quantities in equation (9) (other than Avogadro's number), and he conducts various experiments using different fluids and particles.

4. Each of these measurements, when combined with equation (9),

yields approximately the value 6 10²³ for N.

5. This approximate value for N is also obtained from experiments other than those involving particles suspended in a fluid.

6. From steps 4 and 5 Perrin concludes (“with a probability bordering on certainty”) that molecules exist.

The apparent circularity consists in the fact that in step 1 Perrin is making the crucial assumption that molecules exist. Without this assumption he cannot derive equation (7), which gives a ratio of the number of molecules per unit volume at the height h to the number at the bottom of the cylinder. Is a charge of circularity warranted? In what follows I will consider some attempts to understand Perrin's reasoning so that circularity is avoided.

A. A Common-Cause Interpretation

Wesley Salmon (1984) urges that Perrin's reasoning to the reality of molecules is an example of a legitimate common-cause argument. The

basic idea of such an argument is this. If very similar effects have been produced, and if it can reasonably be argued that none of these effects causes any of the others, then it can be concluded that these effects all result from a common cause.

that, within the accuracy of their experiments, agree with 6 10²³,

we are as impressed by the ‘remarkable agreement' as were Perrin and Poincare. Certainly, these five hypothetical scientists have been count­ing entities that are objectively real” (Salmon 1984, p. 221). He later says, “Remember, for instance, the victims of mushroom poisoning; their common illness arose from the fact that each of them consumed food from a common pot. Similarly, I think, the agreement in values arising from different ascertainments of Avogadro's number results from the fact that in each of the physical procedures mentioned, the experimenter was dealing with substances composed of atoms and molecules—in ac­cordance with the theory of the constitution of matter that we have all come to accept. The historical argument that convinced scientists of the reality of atoms and molecules is, I believe, philosophically impeccable” (Salmon 1984, p. 223).^[151]

Since Salmon claims Perrin's argument is philosophically impeccable, he would deny any circularity charge. According to him, the argument goes like this:

1. If molecules exist, then from experiments on Brownian motion we

get a value for Avogadro’s number of N = 6 10²³.

2. If molecules exist, then from Rutherford’s experiments on

alpha particle decay, we get a similar value for Avogadro’s number. The same is true for experiments involving X-ray diffraction, blackbody radiation, and electrochemical phenomena.

3. There is no reason to suppose that Brownian motion’s resulting in

a value of N = 6 10²³ causes alpha particle decay to yield the

same value, nor vice versa. The same applies to other cases.

4. So probably each phenomenon’s yielding a similar value for N has a common cause, namely, the existence of molecules.

5. So probably molecules exist.

This argument is not circular, since no assumption is made in premises 1 and 2 that molecules do in fact exist. All that is assumed is a conditional: if molecules exist, then.... Salmon himself recognizes this when he writes, “On the basis of my experiments, assuming matter to be composed of mol­ecules, I calculate the number of molecules in a mole of any substance to be.” This corresponds to premises 1 and 2. The problem, however,

is that this conditional assumption is too weak to yield the strong conclu­sion 4. The most that premises 1, 2, and 3 warrant is the conditional

4'. If molecules exist, then probably each phenomenon’s yielding a similar value for N has a common cause.

But 4' is much less than Perrin himself claims. To generate the conclu­sion that Perrin wants, Salmon might alter the argument by adding an additional premise, namely, “Molecules exist.” But now the argument becomes clearly circular. A more promising approach is to delete the antecedent “If molecules exist” from premises 1 and 2 and assert simply that on each of the varied experiments in question physicists calculated the value of N to be 6 10²³, where the latter claim does not presup­pose that this is the correct value or even that molecules exist. On this interpretation, we have similar effects (similar calculations of a number that is supposed to represent a number of molecules); and these effects do not cause one another.

One strategy for doing so would be to argue for two points: (a) that the existence of molecules can cause, or be a causal factor in producing, similar calculations of N from experiments on Brownian motion, alpha particle decay, X-ray diffraction, and so on (this could be done by showing how molecular processes can be involved in, or related to, the phenomena in question); and (b) that other possible causes do not pro­duce these effects. Both before and after giving his common-cause argu­ment involving the five hypothetical scientists, Salmon in fact goes some way toward arguing for points (a) and (b). He considers how N is related to the five phenomena cited. And he discusses one rival to molecular theory, namely, energeticism, which, he argues, is incapable of explain­ing the experimental results. But if indeed it is possible to defend points (a) and (b), then a common-cause argument is both unnecessary and unproductive. It is unnecessary because if (a) and (b) can be successfully defended then the existence of molecules is shown to be probable with­out invoking a common-cause argument. It is unproductive because a common-cause argument does not by itself make probable the existence of molecules, contrary to what Salmon claims is shown by his “five hypo­thetical scientists” argument.

B. A Hypothetico-Deductive Interpretation

A different interpretation is to suppose that Perrin is engaging in a form of hypothetico-deductive reasoning: From the hypothesis that molecules exist and have the properties he attributes to them he draws deduc­tive conclusions regarding observable phenomena, including Brownian motion. He tests these conclusions experimentally and finds they are correct.

From what hypothesis or set of hypotheses is Perrin supposed to have derived observational conclusions, and what observational conclusion(s) does he derive that he takes to confirm the hypothesis? The following hy­pothesis is clearly among those from which Perrin derives consequences:

h. Chemical substances are composed of molecules, the number N of which in a gram molecular weight of a substance is the same for all substances.

A claim (indeed the most important one) that Perrin establishes experi­mentally that he takes to confirm h is this:

C. The calculation of N done by means of Perrin's experiments on Brownian particles, using equation (9), is 6 10²³, and this number remains constant

even when values for n^z, n, and so on, in equation (9) are varied.

Proposition C might well be called “observational.” But it is not something that Perrin derives from his theoretical hypothesis h, nor from h together with other hypotheses he employs about molecules and Brownian par­ticles. What Perrin does is to derive equation (9), not proposition C, de­ductively from such hypotheses. Then he uses equation (9) together with results from various carefully designed experiments to establish C, which he regards as confirming molecular theory.^[154] But this is not the procedure ad­vocated by hypothetico-deductivists. Contrary to the hypothetico-deduc- tive view, the conclusion whose establishment is being claimed to confirm the theory is not derived from that theory.

Even if Perrin does not derive C from his theory, could he have done so? Is C derivable from the theoretical assumptions Perrin in fact makes? No, because even though one of the hypotheses Perrin was using is that N is a constant, he did not begin with any theoretical postulate concerning the numerical value of this constant.

Finally, and perhaps most important, as I will argue in section 4, Per­rin's approach to confirming molecular theory is much richer than that suggested by a hypothetico-deductive approach. He does not in fact defend this theory simply on the grounds that it entails true “observa­tional” conclusions (whether or not these include C). Nor, therefore, is he subject to criticisms of the dubious hupothetico-deductive view of confirmation, according to which if h entails e, then e confirms h (see Achinstein 1983a, chap. 10).

C. Bootstrapping

Clark Glymour's idea of bootstrapping looks more promising than the hypothetico-deductive account because it uses experimental results to­gether with hypotheses in a theory to confirm those hypotheses (Glymour 1980). To invoke Glymour's own simple example, consider the ideal gas law expressed as

where P represents the pressure of a gas, V its volume, T its absolute temperature, and k an undetermined constant. We suppose that we can experimentally determine values for P, V, and T, but not for k. Equation (10) can be “bootstrap confirmed” by experimentally obtaining one set of measurements for P, V, and T and then employing equation (10) itself to compute a value for k. Using this value for k, together with a second set of values for P, V, and T, we can instantiate this equation.

Glymour himself cites Perrin's reasoning in determining a value for Avogadro's number as an example of this type of confirmation (Glymour 1975; in Achinstein 1983b, p. 30, n. 12). Although he does not spell out the Perrin example, presumably what Glymour will say is this: Perrin's equation (9) relates Avogadro's number to measur­able quantities n', n, m, and so on. Using one set of measurements for these quantities, Perrin employed equation (9) itself to compute a value for N. Using this value for N, together with a second set of values for n', n, m, and so on, Perrin instantiated, and thus bootstrap confirmed, equation (9).

In The Nature of Explanation (Achinstein 1983a) I criticize Glymour's general account of bootstrap confirmation on the grounds that it allows the confirmation of equations containing completely undefined or obvi­ously meaningless terms. This objection is related to the point I now want to make.

When the ideal gas equation PV = kT gets confirmed in the manner in­dicated by Glymour—simply by experimentally determining two sets of values for P, V, and T—the term k (at least in Glymour's example) simply represents a constant—that is, a number. This constant can be given a

molecular interpretation.¹¹ But it need not be; it can simply be construed as a constant of proportionality—that is, that number by which T needs to be multiplied to yield the same number as the product PK This is the way that Glymour seems to be treating it. The value of that constant is to be determined experimentally.

Now in Perrin's equation (9) the constant N can be construed in a manner exactly analogous to the way Glymour seems to be treating k in equation (10), that is, as a numerical constant relating the other, physical quantities in equation (9). Indeed, nothing in Glymour's theory of confir­mation requires us to interpret N in equation (9) as a number of anything, let alone a number of molecules.^{[155] [156] (The other quantities in equation [9] are physically interpreted.) Equation (9) would be bootstrap confirmed by two sets of measurements of the quantities n, n', and so on, if N rep­resented the number of angels on the head of a pin, or if N were just like a constant of proportionality. So bootstrap confirming equation (9), or any other equation (e.g., equation [7]) containing the constant N, does not confirm the existence of molecules. (No one, not even Glymour, takes bootstrap confirming the ideal gas equation—equation [10]—to be con­firming the existence of molecules, even though k in equation (10) can be given a molecular interpretation.) But when Perrin determined Avogadro's number from his experiments using equation (9) he took his results to confirm the existence of molecules. Either he was mistaken in doing so, or else Glymour's bootstrap confirmation of equation (9) does not capture, at least not completely enough, the logic of Perrin's reasoning.}

In this section I have noted three ways of construing Perrin's rea­soning to the reality of molecules from his experimental determination of Avogadro's number on the basis of Brownian motion. Salmon's common­cause idea, as he formulates it, is not sufficient or necessary to yield the de­sired conclusion. The hypothetico-deductive account does not adequately represent Perrin's reasoning, since his calculation of N, from which he infers the existence of molecules, is not derived or derivable from the theoretical assumptions he makes. Nor does Glymour's bootstrapping ap­proach to equation (9) permit us to see how Perrin legitimately could have inferred the existence of molecules.

4.