A SOLUTION
I will now suggest a way to understand Perrin's reasoning that avoids circularity and yields an argument free from problems of the previous interpretations.
At the beginning of section 3, I took the first step in Perrin's reasoning to be (in part) this: From various assumptions, including that molecules exist, and that gases containing them satisfy the ideal gas law, Perrin derives equation (7).
This certainly is part of Perrin's reasoning, but it is not his first step. Both in his 1909 article and in his 1913 book, long before he begins to derive equations (7) and (9), he offers a general discussion of the atomic theory and the existence of atoms and molecules making up chemical substances. In his book 82 pages are spent developing atomic theory and giving chemical evidence in its favor before he turns to a discussion of Brownian motion in chapter 3 (Perrin 1990, p. 83). In his article he begins with a description of Brownian motion and then offers arguments that Brownian motion is caused by the agitation of molecules and hence that molecules exist. (This is also how he begins chap. 3 of his book.) Let me briefly mention two such arguments.First, he writes, “It was established by the work of M. Gouy (1888), not only that the hypothesis of molecular agitation gave an admissible explanation of the Brownian movement, but that no other cause of the movement could be imagined, which especially increased the significance of the hypothesis” (Perrin 1984, pp. 510-511). Perrin notes that Gouy's experiments established that known “external” causes of motion in a fluid, including vibrations transmitted to the fluid by external causes, convection currents, and artificial illumination of the fluid, do not produce the Brownian motion. When each of these known causes was reduced or eliminated the Brownian motion continued unabated. So, Perrin concludes, “It was difficult not to believe that these [Brownian] particles simply serve to reveal an internal agitation of the fluid, the better the smaller they [the Brownian particles] are, much as a cork follows better than a large ship the movements of the waves of the sea” (1984, p.
511).Perrin offers a second argument that he regards as even stronger than the first. When a fluid is disturbed, the relative motions of its small but visible parts are irregular. (This can be seen when colored powders are mixed into the fluid.) However, this irregularity of motion does not continue as the visible parts get smaller and smaller. At the level of Brownian motion an equilibrium is established between what Perrin calls “coordination” and “decoordination”: if certain Brownian particles stop, then other Brownian particles in other regions assume the speed and direction of the ones that have stopped. From this Perrin draws the following conclusion:
Since the distribution of motion in a fluid does not progress indefinitely, and is limited by a spontaneous recoordination, it follows that the fluids are themselves composed of granules or molecules, which can assume all possible motions relative to one another, but in the interior of which dissemination of motion is impossible. If such molecules had no existence it is not apparent how there would be any limit to the de-coordination of motion.
On the contrary if they exist, there would be, unceasingly, partial re-coordination; by the passage of one near another, influencing it (it may be by impact or in any other manner), the speeds of these molecules will be continuously modified, in magnitude and direction, and from these same chances it will come about sometimes that neighboring molecules will have concordant motions. (1984, pp. 514-15; italics his)
Perrin is arguing that irregular motions of the parts of a fluid become regular at the level of Brownian particles, in such a way that the total momentum of these particles is conserved. This strongly suggests that the Brownian particles (which are not responsible for their own motion) are being subjected to the influence of smaller particles still—molecules— which exhibit an equilibrium between coordination and de-coordination of motion. He concludes, “In brief the examination of Brownian movement alone suffices to suggest that every fluid is formed of elastic molecules animated by a perpetual motion” (1984, p.
515).In both arguments Perrin appears to be using eliminative-causal reasoning of the following sort:
A. ( 1) Given what is known, the possible causes of effect E (e.g., Brownian motion) are C, C1,..., C (e.g., the motion of molecules, external vibrations, convection currents).
(2) Cp..., C do not produce effect E.
So, probably,
C produces E.
A premise of type 1 may be defended by appeal to the fact that similar known observed effects are produced by and only by one of the types of causes on the list. Alternatively, it may be defended by appeal to more general established principles mandating one of these causes for an effect of that type. (Such principles may also provide a mechanism by means of which E can be produced by one or more of these causes.) A premise of type 2 may be defended by appeal to the fact that effect E is achieved in the absence of C1,..., Cn, or even when these causes are varied. In the first argument, for example, Perrin claims that Gouy's experiments take into account known external causes of motion in a fluid, and that molecular motion can in principle produce Brownian motion, since “the incessant movements of the [postulated] molecules of the fluid, which striking unceasingly the observed [Brownian] particles, drive about these particles irregularly through the fluid, except in the case where these impacts exactly counterbalance one another” (1984, p. 513). This causal possibility, he obviously believes, can be defended by appeal to more general mechanical principles.[157] In addition, in the first argument Perrin defends premise 2 by appeal to the fact that the motion of the Brownian particles exists whether or not there is external agitation, convection currents, or the like.
Although causation is invoked in these arguments, this is different from Salmon's version of the common-cause argument. For one thing, a common cause of different phenomena is not inferred here. For another, in a common-cause argument no premises of types 1 and 2 need appear.
Nor do they in Salmon's “five hypothetical scientists” argument cited earlier. (Nor are they required by the conjunctive fork conditions; see n. 9.)Even if arguments of type A do not establish the existence of molecules with certainty—since other possible causes cannot be precluded with certainty—Perrin believes that his arguments make it likely that Brownian motion is caused by the motion of molecules that make up the fluid. Accordingly, before any discussion of his own experimental results leading to his determination of Avogadro's number, and then to his claim that molecules exist, Perrin presents preliminary reasons to believe that latter claim. (In his article, as well as in his book, chemical arguments are also presented, e.g., from combinations of elements and compounds.) In addition, he presents reasons for believing that Avogadro's number exists—that is, that the number of molecules in a substance whose weight in grams is its molecular weight is the same for all substances (1984, pp. 515-516; 1990, pp. 18ff.). Finally, as Perrin notes, values for Avogadro's number determined by experiments on phenomena other than Brownian motion yield approximately 6 o 1023 (1990, pp. 105, 215; 1984, pp. 521-524, 583-598). Accordingly, independently of his own experimental results with granules in an emulsion, Perrin clearly believed that there was information (call it background information b) available to him and to other physicists and chemists that supported the following theoretical proposition:
T Chemical substances are composed of molecules, the number N of which in a gram molecular weight of any substance is (approximately) 6 o 1023.[158]
Perrin believed that the probability of T on information b alone was reasonably high, or, using probability notation, he believed that
where k represents some threshold of high probability. He gives arguments (some noted previously) to support claim (i)
Now, as indicated earlier, the experimental result achieved by Perrin on the basis of which he most firmly concludes that molecules exist is
C.
The calculation of N done by means of Perrin’s experiments on Brownian particles, using equation (9), is 6 o 1023, and this number remains constant even when values for n, n, and so on, in equation (9) are varied.Proposition C is not a deductive consequence of T. Even if T is true, C could be false, since the particular experimental assumptions and conditions introduced by Perrin are not required by T to be appropriate to test T. To be sure, we might add experimental assumptions to T that would yield C as a deductive consequence. Such assumptions would include that the Brownian particles of gamboge employed by Perrin all have the same mass and volume, that such particles can be treated like large molecules obeying the standard gas laws, and that Stokes's law is applicable to Brownian particles. Perrin gave empirical arguments for each of these and other assumptions he made.[159] [160] [161] But suppose that instead of adding these assumptions to T, we simply add to the background information b Perrin's experimental results which do not deductively entail these assumptions but (Perrin believed) make them probable. If so, then, even if T together with all the information we are now counting as part of the background information b (including other determinations of N) does not entail the experimental result C, it does make C probable. At least C becomes more probable given the truth of T than without it. That is, it is more likely that Perrin's experiments will yield N = 6 1023 given the assumption that there are molecules whose number N = 6 1023 than it is without such an assumption. So The following is a theorem of probability (proof in appendix): Theorem (iii) states that if T increases C's probability on b, and if both T's and C's probabilities on b are greater than zero, then C increases T's probability on b. Now, from claim (i), T's probability on b is not zero. And, since b contains the information that other experimental determinations of N yield approximately 6 1023, C's probability on b is also not zero. So, it follows from claim (ii) and theorem (iii) that Perrin's experimental result C increases the probability of the theoretical assumption T, that is, Finally, if T's probability on b alone is high—that is, if claim (i) is true— then we can conclude that T's probability on C&b is at least as high, and hence that Result (iv) will be of interest to those philosophers of science who consider increase in probability as sufficient (as well as necessary) for evidence. According to a standard view, On this conception of evidence, Perrin's experimental result C concerning the calculation of Avogadro's number from experiments on Brownian motion counts as evidence in favor of his theoretical assumption T, which postulates the existence of molecules. Moreover, the greater the boost in probability that T gives to C in claim (ii), the greater the boost in probability that C gives to T in result (iv). Assuming these boosts in probability are high, on the standard conception of evidence expressed in proposition (11), we can conclude that Perrin's experimental result C provides strong evidence for his theoretical assumption T (In section 5, I discuss Perrin's reasons for believing that his evidence was so strong.) Elsewhere (see chapter 1), I criticize the increase-in-probability account of evidence (proposition [11]), arguing that it provides neither a necessary nor a sufficient condition for evidence. In its place I advocate this definition (PE) e is potential evidence that h, given b, if and only if (a) e and b are true; (b) e does not entail h; (c) p(h/e&b) x p (there is an explanatory connection between h and e/h&e&b) > 1.(This is equivalent to p (there is an explanatory connection between h and e/e&b >1). 2J On this conception of evidence, what is important is result (v), not (iv). Increase in probability is neither necessary nor sufficient for evidence. But it follows from condition (c) in (PE) that high probability is necessary. (c) also requires the high probability of an explanatory connection between h and e, given the truth of h&e&b. But this is also satisfied in the case of Perrin. Given T, that chemical substances are composed of molecules, the number N of which in a gram molecular weight of a substance is 6 x 1023, and given C, that the calculation of N from Perrin's experiments using Eq. (9) is 6 x 1023, and given the information noted in b, the probability is high that the reason C obtains is that T is true. The probability is high that Perrin's experimental calculation yielded 6 x 1023 because chemical substances are composed of molecules, the number N of which in a gram molecular weight is 6 x 1023. So (vi) p(there is an explanatory connection between T and C / T & C & b). If, as seems reasonable, we may also suppose that the probabilities in (v) and (vi) are sufficiently high to allow their product to be greater than 2, then condition (c) in (PE) is satisfied. Assuming that C is true, as are the facts reported in b, and that C does not entail T, it follows that Perrin's experimental result C is potential evidence for his theoretical claim T. Since T is true, and since there is an explanatory connection between T and C, the experimental result C is also veridical evidence that T. Perrin's reasoning, so represented, reaches the conclusion that his experimental result from Brownian motion constitutes evidence for the truth of a theoretical claim involving the existence of molecules. It does so on the conceptions of potential and veridical evidence I defend. Moreover, it does so without circularity. There is no undefended assumption at the outset that molecules exist. The claim in (i) that molecules probably exist is based on reasons for their existence cited in the background information b in (i). Perrin does not begin by assuming without argument that molecules probably exist. He begins by providing a basis for this assumption that includes experiments other than the one of concern to him in C.[162] The pitfalls of the three interpretations in the previous section are avoided. Unlike the common-cause and bootstrap accounts, we end up by confirming a claim entailing that molecules exist. Unlike the hypothetico- deductive account, we need not suppose that Perrin's experimental result expressed in C is a deductive consequence of his theoretical assumptions about molecules. Nor need we accept the dubious hypothetico-deductive view of evidence. 5.
