3.1. Specific Heat of Diamond

At the beginning of the twentieth century, a problem with specific heats loomed over parts of the scientific community. The specific heat of a substance is the amount of heat required to raise the temperature of a unit mass of that substance by 1° Kelvin. Lord Kelvin, in a lecture to the Royal Society in 1900, described this problem as a “cloud which has obscured the brilliance of the molecular theory of heat and light during the last quarter of the nineteenth century” (Thomson Reference Thomson1901, 527). Despite many successes of the atomic-molecular theory of gases, the combination of this conception of matter with the Maxwell-Boltzmann doctrine of the equipartition of energy demanded by classical physics yielded mixed results in the prediction of specific heats of various substances. Its application in the Dulong-Petit law successfully predicted the specific heats of certain crystalline structures at high temperatures, but the same law yielded values that were significantly higher than those observed for substances such as diamond at room temperature.Footnote ³ Thus, the observation in question here is the low value for the specific heat of diamond at room temperature.

Before 1906, no one had considered the idea that quantum concepts should be incorporated into an analysis of specific heats or that Planck’s work on blackbody radiation could be relevant for these phenomena. Yet, Einstein motivated his treatment of diamond by arguing that if Planck’s quantum postulate from blackbody radiation goes to the “root of the matter,” then we might well expect other anomalous phenomena in the context of heat and radiation and that such anomalies could potentially be addressed by similar quantum postulates (Einstein Reference ter Haar and Beck1907/1989, 219).

Einstein proposed that the idea of quantization by way of the parameter h be applied to the energy of atoms of solids. He thus required that the energy of an atom oscillating at frequency ν take on values infinitesimally close to the discrete values nhν, where $n=0$, 1, 2… . Using this conjecture in combination with a canonical distribution to describe a system, he calculated the mean energy of an oscillator with frequency ν. By making the simplifying assumption that all the atomic vibrations were independent and of the same frequency ν, Einstein was able to calculate the average energy of one oscillator (Einstein Reference ter Haar1907, 186). The energy of one mole of such a solid would be

(2)$\begin{array}{}& E=3R\frac{h\nu /k}{e^{h\nu /\mathrm{kT}}-1},\end{array}$

and the specific heat can be calculated by differentiating the energy with respect to temperature.

Note that as $\mathrm{kT}/h\nu$ approaches 0, $h\nu /\mathrm{kT}$ grows large, and the expression for E approaches 0 as well. In the limit as $\mathrm{kT}/h\nu$ goes to 1 and larger, the expression takes the value of 3R. As Klein explains, “If the specific heat is plotted as a function of temperature, or rather of ($\mathrm{Tk}/h\nu$), one obtains a curve that rises smoothly and monotonically from zero at the origin and approaches the equipartition value, 3R, asymptotically, when ($\mathrm{Tk}/h\nu$) becomes large” (Reference Klein1965, 176).

This relation predicts that the specific heats of solids that conform reasonably well to Einstein’s assumptions should be very low at low temperatures and will approach the value predicted by classical theories at much higher temperatures. The contradictions between the experimental observations and theoretical predictions based on equipartition occurred primarily among lighter atoms, which could be expected to vibrate at higher frequencies (i.e., higher values of ν) at which the value of E approaches 0. Einstein’s quantized treatment thus fits well with these observations since they would be expected not to conform to the equipartition predictions but to be much lower. In particular, according to the equipartition theorem and the Dulong-Petit law, the molar specific heat of diamond would be 3R. However, the observed value is approximately 0.735R. With their relatively low atomic weights, atoms of carbon could be expected to have a higher oscillation frequency according to Einstein’s assumptions, and thus a lower specific heat than the value predicted by the Dulong-Petit law.

We can see how Einstein’s analysis was able to turn the specific heat of diamond at room temperature into ‘old evidence’ for the quantum postulate. Before this account, there was no explanation for the observed value of this quantity. The quantum postulate put forth above served as Einstein’s starting point. The hypothesis that there was some discretization condition on the energy of the molecules allowed him to describe the specific heats of certain solids in general. The previously observed low value of the specific heat of diamond was then accounted for by this description.

3.2. Balmer’s Formula

The second instance of old evidence I will be considering is the expression of Balmer’s formula for the spectrum of hydrogen and its account in terms of Bohr’s (Reference Bohr1913) presentation of old quantum theory. The primary goal of this work was to provide a preliminary theory of the structure of the atom. The assumption Bohr uses in his model of the hydrogen atom takes the fundamental idea of quantization directly from Planck’s work on radiation phenomena but with a significantly different interpretation. Bohr shows how the application of assumptions found in Planck’s theory of radiation to ideas on the atomic structure of hydrogen results in an account of how electrons might be bound to a positive nucleus in stable states. He considers the behavior of energy in the context of an atomic system and describes the emission process as taking place in quantized amounts, dependent on the parameter h. Using this conjecture along with his elementary atomic model, Bohr was able to account for the Balmer formula, which described the discrete spectral lines observed when hydrogen gas is heated. Our focus here will be on this latter fact.

The Balmer formula can be expressed in a generalized form known as the Rydberg formula,

(3)$\begin{array}{}& \nu =R\left(\frac{1}{{\tau }_2^2}-\frac{1}{{\tau }_1^2}\right),\end{array}$

where R is Rydberg’s constant, and τ₁ and τ₂ are integers greater than or equal to 1 such that ${\tau }_1<{\tau }_2$. These precise spectral lines could not be explained on the classical theory. Indeed, one might say they were doubly anomalous. One would first expect, according to classical electromagnetic theory, that an electron orbiting a nucleus would emit energy proportional to its rotational frequency and that this frequency would change continuously as the energy is emitted, thus yielding energy of a continuous spectrum. Yet, the emitted radiation was of an integral number of specific frequencies, as manifested in a number of discrete lines on the spectrum. Furthermore, even if one were to accept the existence of stable states in the atom corresponding to these discrete spectral lines, classical physics predicts that the lines of the higher harmonics should be sums of the fundamental frequencies, whereas the observed spectral lines were expressed by Ritz in his ‘combination principle’ as differences between the harmonics (Jammer Reference Jammer1966, 69). This made Bohr’s accomplishment even more significant.

Although Balmer’s formula had been identified as accurately describing the observed spectral lines, there was no theoretical account of how and why these lines were produced, until Bohr put forth his model of the hydrogen atom. On this model, electrons moved in stationary orbits that were determined in part by the parameter h. Upon transition from one energy level to another, an electron would emit a discrete amount of energy, hν. Bohr considered the case of a hydrogen atom, which was generally accepted as a system in which a single electron rotated around a positive nucleus of charge e. According to his earlier calculations, the binding of an electron to a positive nucleus in a transition to that stable state would result in a certain amount of emitted energy.

To express the amount of energy emitted by the system when transitioning from a state τ₁ to another state τ₂, we have

(4)$\begin{array}{}& W_{{\tau }_2}-W_{{\tau }_1}=\frac{2{\pi }^2{\mathrm{me}}^4}{h^2}\left(\frac{1}{{\tau }_2^2}-\frac{1}{{\tau }_1^2}\right),\end{array}$

where m and e are the mass and charge of an electron. By comparing with equation (3), we see that this is simply an expression of the formula put forth by Balmer, where R is replaced by more specific constants.

Thus, we see that Bohr was able to use a quantum postulate to construct an atomic model that was able to account for a previously observed phenomenon.Footnote ⁴ By using a quantum postulate to formulate a model of the hydrogen atom, and then using this model to calculate correct values for the previously observed spectral lines of hydrogen that had no explanation, Bohr turned those observations into ‘old evidence’ for the quantum postulate.

3.3. Bayesian Analysis

Although the conditions given by Hartmann and Fitelson listed above are meant to be plausible enough to apply in any case, it is useful to consider what they mean in our particular context. We have seen that the conditions are all based on comparative probabilities of a hypothesis H, conditional on various combinations of ways that a phenomenon might be accounted for by H or by its best alternative H′. Thus, in this context, we should read H, X, and Y as expressing claims about the quantum postulate (QP) in relation to the phenomenon of specific heat of diamond or Balmer’s formula. Specifically, X and Y are as follows.

While I conduct my analysis in terms of specific heats, the same points apply, mutatis mutandis, to the account of Balmer’s formula by Bohr’s old quantum theory.

Recall that Hartmann and Fitelson’s result requires the acceptance of four conditions, as outlined in section 2. In our case, the first condition, which can be written $\mathrm{Pr}\left(\mathrm{QP}\right|X\text{\&}\neg Y)>\mathrm{Pr}(\mathrm{QP}|\neg X\text{\&}\neg Y)$, states that the quantum postulate is more likely if it is able to account for the specific heat of diamond and classical physics is not than if neither it nor classical physics with its assumption of continuity could account for this phenomenon. Historically, it was this ability to explain the phenomenon that rendered it a promising postulate for many scientists, so the ordinal ranking is reasonable. The second condition says that QP is more likely if it accounts for the specific heat of diamond while classical theory does not, as opposed to a situation in which classical theory could account for it and a quantum postulate could not. This reflects the idea that if classical theory could account for the specific heat values and QP could not, there would have been no reason to think of QP as being promising in relation to this phenomenon. Indeed, this condition was expressly satisfied historically, since it did indeed seem that classical physics could not account for the specific heat of diamond, while a quantum postulate could.

The third condition says that QP is less likely in the case that only classical theory could explain the specific heat than in the case in which both QP and classical theory are possible explanations of the specific heat. This seems reasonable, since even if classical theory were able to account for the specific heat, the ability of QP to do so as well seems like it would render QP more promising than if it could not. One might claim that if classical theory could account for the specific heat of diamond, then scientists would not have been concerned with a quantum postulate in this context at all. While this may very well be true, this condition simply ranks credence in the quantum postulate as being higher if it accounts for the specific heat than if it does not; the actual values may well be very low. The last condition states that QP is at least as likely if both it and its competitor explain the specific heat than if no theory at all does so. Again, this seems reasonable: imagine that at the same time that QP was shown to be able to account for the specific heat, a plausible way of adjusting classical theories to account for the phenomenon was also suggested. One might not think that QP is extremely likely in this case, but its ability to account for the phenomenon should not make it less likely than previously.

Given these assumptions, Hartmann and Fitelson show that $\mathrm{Pr}\left(\mathrm{QP}\right|X)>\mathrm{Pr}(\mathrm{QP})$. This means that, if one were to learn that QP accounts for the specific heat of diamond, then this raises one’s degree of belief in QP. Similarly, learning that QP accounts for Balmer’s formula raises one’s degree of belief in QP. This particular solution is fitting for a historical reconstruction for other reasons. The first is the use of a quite general idea that a theory “accounts for” a phenomenon. Hartmann and Fitelson purposely allow for an open interpretation as to what this means, which makes it particularly suitable for an explication of the role of the quantum postulate. Interpretations of what QP actually represented physically were quite varied, but this fact is not a problem in this framework. The second reason is the contention that what is being learned that contributes to the support for QP is the fact that it accounts for a phenomenon that was previously known. This certainly tracks what scientists seemed to be motivated by historically, since the interest always arose in response to a demonstration that the quantum postulate could account for the phenomena. We thus see how Hartmann and Fitelson’s solution provides an accurate description of historical results for these phenomena.

5.1. Surprising Application

First, consider the difference between $\mathrm{Pr}\left(\mathrm{QP}|X\right)$ and Pr(QP), which can be written as follows:

(5)$\begin{array}{}& \mathrm{Pr}\left(\mathrm{QP}\right|X)-\mathrm{Pr}(\mathrm{QP})=\mathrm{Pr}(\neg X\left)\right[\mathrm{Pr}\left(\mathrm{QP}\right|X)-\mathrm{Pr}(\mathrm{QP}|\neg X)].\end{array}$

Let us suppose that the conditional probabilities for QP on X and ¬X are fixed, regardless of the value of Pr(X). That is, we suppose that regardless of how likely we think it is that the quantum postulate accounts for the specific heat of diamond, the fact of its accounting for this phenomenon has a fixed relation with our credence in the postulate itself. Then the amount to which one’s degree of belief in QP is raised is proportional to Pr(¬X). In other words, it is a decreasing function of Pr(X), one’s prior belief that the hypothesis accounts for the phenomenon. This means that if before learning definitely that QP accounts for the specific heat one already thought that this was likely to be the case, then learning this fact does not add as much to the support for QP as if one were more skeptical of the idea that QP could account for this phenomenon.

In our particular case, it does seem as though it would be relatively surprising that QP was able to account for the specific heat of diamond. This is, in part, due to the fact that the QP itself was not obviously reconcilable with certain successful aspects of classical physics. What made it even more surprising was its application to a context that seemed unrelated to any of the contexts in which QP was previously successful, that is, to a theory of energy in solids rather than the context of energy transmission between radiation and matter. Thus, it seems reasonable that Einstein’s demonstration that a quantum postulate could account for aspects of the specific heat of diamond played a significant role in raising scientists’ degree of belief in this hypothesis, given its very surprising nature. It is worth noting here that this is different from the idea of the evidence itself being surprising: what is in question is not the result itself but the ability of a particular (quantum) hypothesis to account for the evidence in question.

5.2. Anomalous Evidence

I now turn to the significance of anomalous phenomena I discussed in section 4. While Hartmann and Fitelson do not discuss this issue, I drew attention to the fact that, historically, it seems that learning that a hypothesis accounts for a previously known phenomenon is not always taken to be strong evidence for that hypothesis. The cases in which we put the most weight on old evidence are those in which it was believed that this evidence was not accounted for by any previous theory. In the classic example of general relativity being able to account for the precession of the perihelion of Mercury, it was intuitively seen as highly significant that Einstein’s theory could account for this phenomenon, whereas being able to account for the observations that Newton’s theory also covered would be merely expected of any competing theory. In the cases of both specific heats of diamond at room temperature and the spectral lines expressed by Balmer’s formula, it seemed significant that classical theory yielded predictions that were in direct conflict with observations of these phenomena.

I take it that this fact can serve as a guide to conditions that a solution to old evidence should satisfy. We can express this as the condition that an agent’s credence in a hypothesis should be boosted the most in cases in which it can account for a phenomenon that previous theories could not account for. In other words, the support for H (or in our case QP) should vary with an agent’s credence in Pr(Y), that is, how likely we think it is that the classical theory can account for the phenomena in question. If we examine Hartmann and Fitelson’s framework, we can infer that such a correlation holds in the case in which two separate conditions are satisfied: first, the relation $a-c>b-d$ holds, and second, Pr(Y) and Pr(X) are independent. When the latter condition holds, $\mathrm{Pr}(\neg Y|X)=\mathrm{Pr}(\neg Y|\neg X)$, so that $x=y$. When this is true, we see that equation (1) becomes $\mathrm{ax}+b(1-x)>\mathrm{cx}+d(1-x)$. If the first condition holds, the difference between the two sides of the equation varies with x, as desired. I argue that these conditions are satisfied in cases we would intuitively consider accounting for old evidence to have contributed to an agent’s credence in a hypothesis.

First consider the assumption that Pr(Y) and Pr(X) are independent. The Pr(Y) refers to the credence an agent has in the proposition that classical theory accounts for the specific heat of diamond at room temperature, while Pr(X) is the credence an agent has in the proposition that a quantum postulate accounts for the specific heat of diamond at room temperature. Before the quantum postulate was ever applied to this phenomenon, it was known that classical theory could not account for the specific heat, and so Pr(Y) was very low. The subsequent use of the quantum postulate had no bearing on this fact. Thus, it is reasonable to think that at least in this case, these two probabilities are independent. Similarly, in the case of Balmer’s formula, Bohr’s introduction of old quantum theory to provide an account of this formula did not change a low credence in the proposition that classical theory could account for these spectral lines.

One might be concerned about this condition since the relations between Pr(X) and Pr(Y) are features of the probability space. Thus, there is perhaps simply a fact of the matter that a low credence in Y would affect the credence in X, and so we cannot assume independence. However, in the situation we are considering, we have explicitly eliminated the assumption of logical omniscience of the agent. This means that we can assign values to Pr(X) and Pr(Y) that make sense for an agent in that context, and at the time point that we are considering, it is reasonable to consider the two credences as being independent. After all, even if it was known that the classical theories did not account for the phenomena, the quantum postulate modifies a very specific aspect of classical theories. Presumably, there are countless other ways to modify classical theory that would result in hypotheses that would not account for the specific heat. Of course, this independence condition will not always hold: for instance, after the formulation of Bohr’s correspondence rule, the ability of QP to account for phenomena in certain domains would be known to depend on the ability of certain classical descriptions of phenomena. Such relations would have to be evaluated with the individual cases in mind.

Now consider the condition that $a-c>b-d$. This is the same as a condition that $a-b>c-d$. The difference $a-b$ is the difference between $\mathrm{Pr}\left(H|X\text{\&}\neg Y\right)$ and $\mathrm{Pr}\left(H|X\text{\&}Y\right)$. The difference $c-d$ is the difference between $\mathrm{Pr}\left(H|\neg X\text{\&}\neg Y\right)$ and $\mathrm{Pr}\left(H|\neg X\text{\&}Y\right)$. I claim that it is reasonable to expect the first probability difference to be larger than the second.

This is because in the first case, we are assuming that H accounts for a phenomenon, and the difference relies on whether there is another hypothesis that does the same. In the second case, we are evaluating the probability of a hypothesis conditional on the fact that the hypothesis does not account for the phenomenon in question. I would argue that the presence of an alternative explanation is more meaningful in the first case. In the first case, when a hypothesis accounts for evidence, a reasonable thing to ask is whether there are alternative explanations for the phenomenon. In our particular example, this expresses the idea that the quantum postulate’s ability to account for the specific heat of diamond when classical theory could not was significant. However, if a hypothesis does not account for the evidence in question, the latter fact seems to be less important, in the sense that our credence in H was likely to be lower in the first place.

Imagine, for instance, that we are assigning a credence to the old quantum theory, conditional on the fact that it does not account for the observation of certain spectral lines, such as those of helium. This failure to account for a given phenomenon seems to render our credence in that postulate quite low. This is still true when the best possible competitor to H also fails to explain the phenomenon. Imagine now that a theory is developed that could account for the spectral lines. It is true that this might lower our credence in old quantum theory even more, but given the latter’s failure to account for the phenomenon in which we were originally interested, there is less of a margin by which Pr(H) may be reduced. In our particular case, credence in a quantum postulate that did not explain the specific heat would be low, regardless of whether classical theory could account for specific heat. Therefore, we would expect that the difference between c and d is smaller than the difference between a and b.

When the two above conditions are satisfied, the difference between $\mathrm{Pr}\left(H|X\right)$ and Pr(H) will vary with Pr(¬Y). Particular numbers will depend of course on the priors an agent assigns to all the conditional probabilities. However, this dependence captures the fact that accounting for something like specific heat, where Pr(¬Y) is very high, will provide relatively more support for QP than accounting for a phenomenon like Stokes’s rule, where Pr(¬Y) is considerably lower.

5.3. Discussion

Of course, the formal conditions that I have introduced may not always be satisfied in historical cases of old evidence. For instance, it might be the case that an agent knows the precise ways in which Pr(Y) and Pr(X) are probabilistically related, and so independence between those will not hold. This would simply mean that the feature I identify here does not play a role in the explication of the support of old evidence. Furthermore, this is not to claim that a phenomenon such as Stokes’s rule does not provide any support to the quantum postulate but merely to claim that the support is less in comparison to the other phenomena.

Finally, we can see the significance of these results for Bayesian approaches in general due to an important reflection of a feature that is a standard result of Bayesian confirmation theory. We know that if evidence E is a logical consequence of hypothesis H, the support E provides to H is higher when it is more surprising or unexpected (i.e., when Pr(E) is lower). Thus, if one were to analyze the logical relations between the examples given here, one could say that the evidential result of the low specific heat of diamond better confirms the quantum postulate than Stokes’s rule because we should assign a lower prior probability to this evidential result that seems to conflict with classical theory. Consequently, we need not appeal to old evidence or anomalous phenomena at all to explain this difference.Footnote ⁵ However, I would argue that this result actually bolsters the importance of the analysis given here, in the following sense.

Note that Hartmann and Fitelson’s solution to POE aims essentially to provide an accurate description of how old evidence for a hypothesis can affect an agent’s credence in that hypothesis. By introducing extrasystematic statements such as X and Y, Hartmann and Fitelson explicitly consider an agent’s credal state when learning something about the relation between a theory and previously known observations because the logical relations that exist between theory and evidence were not accessible before this point. However, it is desirable that a descriptive account of an agent’s reasoning at this point corresponds to results in Bayesian confirmation that match with our methodological intuitions. What I have shown is that the framework Hartmann and Fitelson give for explicating the role of old evidence when $\mathrm{Pr}\left(E\right)=1$ corresponds to the intuitively plausible result of a purely logical analysis of the relations between the hypothesis and evidential phenomena. Given that the historical case I have presented is exactly one in which $\mathrm{Pr}\left(E\right)=1$ for each of the phenomena in question, this analysis provides an account of how scientists may have made the judgments they did in a way that corresponds to the logical relation.