So far, the probabilistic NMA only took the predictive and explanatory success of T as evidence for the realist position. But perhaps, different kinds of evidence bear on the argument, too. For example, Ludwig Fahrbach (2009, 2011) has argued that the stability of scientific theories in recent decades favors scientific realism. In this section, we show how such arguments could be part of a probabilistic NMA. We do not want to take a stand on the historical correctness of Fahrbach’s observations: rather, we would like to demonstrate how such observations can in principle affect the NMA.
Fahrbach’s argument is based on scientometric data. He observes an exponential growth of scientific activity, with a doubling of scientific output every 20 years (Meadows 1974). He also notes that at least 80 % of all scientific work has been done since the year 1950 and observes that our fundamental scientific theories (e.g., the periodic table of elements, optical and acoustic theories, the theory of evolution, etc.) were stable during that period of time. That is, they did not undergo rejection or major conceptual change. Laudan’s examples in favor of PMI, on the other hand, all stem from the early periods of science: the caloric theory of heat, the ether theory in physics, or the humoral theory in medicine.
For giving a fair assessment of PMI, we have to take into account the amount of scientific work done in a particular period. This implies, for example, that the period 1800–1820 should receive much less weight than the period 1950–1970. According to Fahrbach, PMI fails because most “theory changes occurred during the time of the first 5 % of all scientific work ever done by scientists” (Fahrbach 2011, 149). If PMI were valid, we should have observed more substantial theory changes or scientific revolutions in the recent past. However, although the theories of modern science often encounter difficulties, revolutionary turnovers do not (or only very rarely) happen. According to Fahrbach, PMI stands refuted—or at the very least, it is not more rational than optimistic meta-induction.
The factual correctness of Fahrbach’s observations may be disputed, and his model is certainly very simplified. Yet, it deserves to be taken seriously, particularly with respect to the implications for the NMA. In this paper, I explore if observations of long-term stability expand the set of circumstances where the NMA holds. To this end, let us refine our probabilistic model of the NMA.
As before, the propositional variable H expresses the empirical adequacy of theory T, and S denotes the predictive, retrodictive and explanatory success of T. Similar to Dawid et al. (2015), we introduce a integer-valued random variable A that expresses the number of satisfactory alternatives to T, including unconceived theories. In the individuation of alternatives, we stick with the Dawid et al. paper: that is, we demand that alternative theories satisfy a set of (context-dependent) theoretical constraints \(\mathcal {C}\), are consistent with the currently available data \(\mathcal {D}\), and give distinguishable predictions for the outcome of some set \(\mathcal {E}\) of future experiments. In line with our focus on empirical adequacy rather than truth, we do not distinguish between empirically equivalent theories with different theoretical structures. Finally, major theory change in the domain of T is denoted by variable C, and absence of change and theoretical stability by ¬C. “Theory change” is understood in a broad sense, including scenarios where rivalling theories emerge and end up co-existing with T.
The dependency between these four propositional variables—A, C, H and S—is given by the Bayesian network in Fig. 4. S, the success of theory T, only depends on the empirical adequacy of T, that is, on H. The probability of H depends on the number of distinct alternatives that are also consistent with the current data, etc. Finally, C, the probability of observing substantial theory change, depends on S and A: the empirical success of T and the number of available alternatives to T that scientists might develop. H affects C only via A. All this is captured in the following assumption:
-
A0
The variables A, C, H and S satisfy the conditional independencies in the Bayesian Network of Fig. 4.
To rule out preservation of a theory by a of series degenerative accommodating moves, the variable C should be evaluated over a longer period (e.g., 30–50 years). We now define a number of real-valued variables in order to facilitate calculations:
-
Denote by a
j
:=p(A=j) the probability that there are exactly j (not necessarily discovered) alternatives to T that satisfy the theoretical constraints \(\mathcal {C}\), are consistent with current data \(\mathcal {D}\) and give definite predictions for future experiments \(\mathcal {E}\), etc.Footnote 3
-
Denote by h
j
:=p(H|A=j) the probability that T is empirically adequate if there are exactly j alternatives to T.
-
As before, denote by s:=p(S|H) and s
′:=p(S|¬H) the probability that T is successful if it is (not) empirically adequate.
-
Denote by c
j
:=p(¬C|A=j,S) the probability that no substantial theory change occurs if T is successful and there are exactly j alternatives for T.
Suppose that we now observe ¬C (no substantial theory change has occurred in the last decades) and S (theory T is successful). The Bayesian network structure allows for a simple calculation of the posterior probability of H:
$$\begin{array}{@{}rcl@{}} p(\neg CSH) &=& \sum\limits_{A} p(A) p(\neg C|AS) p(S|H) p(H|A) \\ &=& \sum\limits_{j=0}^{\infty} a_{j} \, c_{j} \, s \, h_{j} \\ p(\neg CS) &=& {\sum}_{A, H} p(A) p(\neg C|AS) p(S|H) p(H|A) \\ &=& \sum\limits_{A} p(A) p(\neg C|AS) p(S|H) p(H|A) + \sum\limits_{A} p(A) p(\neg C|AS) p(S|\neg H) p(\neg H|A) \\ &=& \sum\limits_{j=0}^{\infty} a_{j} \, c_{j} \, (s \, h_{j} + s^{\prime} (1-h_{j})) \end{array} $$
With the help of Bayes’ Theorem, these equations allows us to calculate the posterior probability of H conditional on C and S. That is, how probable is H given that T is successful and that we have observed no major theoretical change in recent decades?
$$ p(H|\neg CS) = \frac{p(\neg CSH)}{p(\neg CS)} = \frac{\sum\nolimits_{j=0}^{\infty} a_{j} \, c_{j} \, s \, h_{j}}{\sum\nolimits_{j=0}^{\infty} a_{j} \, c_{j} \, (s \, h_{j} + s^{\prime} (1-h_{j}))} $$
(3)
We now make some assumptions on the values of these quantities.
-
A1
If T is empirically adequate then it will be successful in the long run: s=p(S|H)=1.Footnote 4
-
A2
The empirical adequacy of T is no more or less probable than the empirical adequacy of an alternative which satisfies the same set of theoretical and empirical constraints: h
j
:=p(H|A
j
)=1/(j+1). Scientists are indifferent between theories which satisfy the same set of theoretical and empirical constraints.
-
A3
The more satisfactory alternatives are in principle accessible to the scientists, the less likely is an extended period of theoretical stability. In other words, c
j
:=p(¬C|A=j,S) is a decreasing function of j. For convenience, we choose c
j
=1/(j + 1).
While A2 is a default assumption made for expositional ease, A3 is more substantive: ceteris paribus, the probability of long-term theoretical stability decreases the more satisfactory alternatives for a successful theory exist. This qualitative claim is very intuitive. More contentious is the precise rate of decline. That’s why we will relax the peculiar assignment of the c
j
later on in the paper.
-
A4
Assume that T is our currently best theory and we happen to find a satisfactory alternative T
′. Then, the probability of finding yet another alternative T
″ is the same as the probability of finding T
′ in the first place. Formally:
$$ p(A>j|A \ge j) = p(A>j+1|A \ge j+1) \; \forall j \ge 0. $$
(4)
In other words, Eq. 4 expresses the idea that the number of known alternatives does not, in itself, raise or lower the probability of finding another alternative.
Note that A0–A4 are equally plausible for the realist and the anti-realist. In other words, no realist bias has been incorporated into the assumptions. We can now show the following proposition (proof in the Appendix):
Proposition 1
If a
0
>0, then A4 is equivalent to the requirement a
j
:=a
0
⋅(1−a
0
)
j.
Together with this proposition, A0–A4 allow us to rewrite Eq. 3 as follows:
$$ p(H|\neg CS) = \frac{\sum\nolimits_{j=0}^{\infty} (1-a_{0})^{j} \, \frac{1}{(j+1)^{2}}} {\sum\nolimits_{j=0}^{\infty} (1-a_{0})^{j} \, \frac{1-s^{\prime}j}{(j+1)^{2}}} $$
(5)
With the help of this formula, we can now rehearse the NMA once more and determine its scope, that is, those parameter values where p(H|¬C
S)>1/2. The two relevant parameters are a
0, the probability that there are no satisfactory alternatives to T, and s
′, the probability that T is successful if it is not empirically adequate. Since an analytical solution of Eq. 5 is not feasible, we conduct a numerical analysis. Results are plotted in Fig. 5.
These results are very different from the ones in Section 2. With the hyperplane z=0.5 dividing the graph into a region where T may be accepted and a region where this is not the case, we see that the scope of the NMA has increased substantially compared to Fig. 3. For instance, a
0>0.1 suffices for a posterior probability greater than 1/2, even if the value of s
′ is very high. This is a striking difference to the previous analysis where way more optimistic values had to be assumed in order to make the NMA work.
So far, the analysis has been conducted in terms of absolute confirmation, that is, the posterior probability of H. We now complement it by an analysis in terms of relative or incremental confirmation. That is, we calculate the degree of confirmation that ¬C
S exerts on H. We use four different measures to show the robustness of our results. First, we calculate the log-likelihood measure l(H,E)= log2p(E|H)/p(E|¬H) which has a good reputation in formal theories of evidence (e.g., Hacking 1965; Fitelson 2001) and a firm standing in scientific practice (e.g., Royall 1997; Good 2009). To apply it to the present case, we calculate
$$\begin{array}{@{}rcl@{}} p(\neg CS|H) = \frac{p(\neg CSH)}{p(H)} &=& \frac{{\sum}_{A} p(A) p(\neg C|AS) p(S|H) p(H|A)}{{\sum}_{A} p(A) \, p(H|A)} \\ &=& \frac{\sum\nolimits_{j=0}^{\infty} a_{j} \, c_{j} \, s \, h_{j}}{\sum\nolimits_{j=0}^{\infty} a_{j} \, h_{j}} \\ &=& \frac{{\sum}_{j=0}^{\infty} (1-a_{0})^{j} \frac{1}{(1+j)^{2}}}{\sum\nolimits_{j=0}^{\infty} (1-a_{0})^{j} \frac{1}{1+j}} \end{array} $$
$$\begin{array}{@{}rcl@{}} p(\neg CS|\neg H) = \frac{p(\neg CS \neg H)}{p(\neg H)} &=& \frac{\sum\nolimits_{A} p(A) p(\neg C|AS) p(S|\neg H) p(\neg H|A)}{\sum\nolimits_{A} p(A) \, p(\neg H|A)} \\ &=& \frac{\sum\nolimits_{j=0}^{\infty} a_{j} \, c_{j} \, s^{\prime} \, (1-h_{j})}{\sum\nolimits_{j=0}^{\infty} a_{j} \, (1-h_{j})} \\ &=& \frac{\sum\nolimits_{j=0}^{\infty} (1-a_{0})^{j} \frac{s^{\prime}j}{(1+j)^{2}}}{\sum\nolimits_{j=0}^{\infty} (1-a_{0})^{j} \frac{j}{1+j}} \end{array} $$
l(H,E) is a confirmation measure that describes the discriminative power of the evidence with respect to the realist and the anti-realist hypothesis. It is relatively insensitive to prior probabilities. Other measures aim at quantifying the increase in degree of belief from p(H) to p(H|E). Typical representatives of that class are the log-ratio measure r(H,E), the difference measure d(H,E) and the Crupi-Tentori measure z(H,E). The definitions of all measures are given below (see also Crupi 2013), and their values follow straightforwardly from the above calculations.
$$\begin{array}{@{}rcl@{}} l(H, E) &=& \log_{2} \frac{p(E|H)}{p(E|\neg H)} \quad\,\,\,\,{\kern.5pt} r(H, E) = \log_{2} \frac{p(H|E)}{p(H)} \\ d(H, E) &=& p(H|E) - p(H) \quad z(H, E) = \frac{p(H|E) - p(H)}{1-p(H)} \;\; \text{(if \(p(H|E) > p(H)\))} \end{array} $$
To deliver a comprehensive picture and to show the robustness of our claims, we calculate the degree of confirmation for all four confirmation measures. All measures are normalized such that a value of zero corresponds to neither confirmation nor disconfirmation. In addition, d(H,E) and z(H,E) only take values between -1 and 1.
Figure 6 plots the degree of confirmation as a function of the value of s
′, for three different values of a
0, namely 0.01, 0.05 and 0.1. As visible from the graph, the degree of confirmation is substantial for all four measures. In particular, it is robust vis-à-vis the values of a
0 and s
′, contrary to the anti-realist argument from Section 2. For example, if s
′ is quite small, as it will often be the case in practice, then l(H,¬C
S) comes close to 10, which corresponds to a ratio of more than 1,000 between p(¬C
S|H) and p(¬C
S|¬H)! This finding accounts for the realist intuition that the stability of scientific theories over time, together with their empirical success, is strong evidence for their empirical adequacy.
Finally, we conduct a robustness analysis regarding A3. Arguably, the function c
j
:=p(¬C|A=j,S)=1/(j+1) suggests that scientists are quite ready to give up on their currently best theory in favor of a good alternative. But as many have philosophers and historians of science have argued (e.g., Kuhn 1977), scientists may be more conservative and prefer the traditional framework, even if good alternatives exist. Therefore we also analyze a different choice of the c
j
, namely \(c_{j} := e^{-\frac {1}{2} \left (\frac {j}{\alpha } \right )^{2}}\), where c
j
falls more gently in j. This leads to the following expression for the posterior probability of H:
$$p(H|\neg CS) = \frac{{\sum}_{j=0}^{\infty} (1-a_{0})^{j} \, \frac{1}{j+1} e^{-\frac{1}{2} \left( \frac{j}{\alpha} \right)^{2}}} {{\sum}_{j=0}^{\infty} (1-a_{0})^{j} \, \frac{1-s^{\prime}j}{j+1} \, e^{-\frac{1}{2} \left( \frac{j}{\alpha} \right)^{2}}} $$
The graph of p(H|¬C
S), as a function of a
0 and s
′, is presented in Fig. 7. We have set α=4, which corresponds to a gradual decline of c
j
. Yet, the results match those from Fig. 5: the scope of the NMA is much larger than in the simple version of the probabilistic NMA. Hence, our findings seem to be robust toward different choices of c
j
. To achieve a result similar to the one for the primitive NMA (Fig. 3), one would have to choose α=8, which will only be the case if scientists are really reluctant to reject the currently best theory. In such circumstances, stability is the default state of a discipline and will not strongly support the NMA.
All in all, our model shows that a probabilistic NMA need not be doomed. Its validity depends crucially on the properties of the disciplinary context where it operates in. This involves the existence of satisfactory alternatives and whether or not the discipline has been in a long period of theoretical stability. Of course, our model makes simplifying assumptions, but unlike those in Section 2, they do not carry a realist bias. This allows for a more nuanced and context-sensitive assessment of realist claims.