1. A Model of Scientific Inquiry

To distinguish between and make precise several formulations of the Independence Thesis, we introduce the following idealized model of scientific inquiry.Footnote ⁷ We describe features of our model fairly informally; technical details can be found in Mayo-Wilson, Zollman, and Danks (Reference Mayo-Wilson, Zollman and Danks2010). In our model, there is a finite collection of scientists, whom we will also call learners. Each scientist has access to the research being conducted by a set of his or her peers, whom we will call neighbors. A scientist’s neighborhood can be understood as her closest colleagues: those researchers with whom she communicates and exchanges information, papers, draft papers, and so on. We assume that the sharing of information is symmetric, in the sense that if scientist 1 knows about scientist 2’s research, then 2 knows about 1’s research.

We represent the relationships in a scientific community using undirected graphs like the one in figure 1. We will often refer to such a community as a research network or simply a network. For simplicity, we assume that the network structure is fixed for the duration of inquiry, although real-world scientific communities are of course dynamic and change over time. Finally, we assume that our research networks are connected: there is an “informational path” (i.e., a sequence of edges) connecting any two scientists. An informational path represents a chain of scientists who each communicate with the next in the chain. In this model, one learns about the research results of one’s neighbors only; individuals who are more distant in the network may influence each other but only by influencing the individuals between them.Footnote ⁸ We restrict ourselves to connected networks because unconnected research networks do not adequately capture scientific communities but rather several separate communities. Examples of a connected and an unconnected network are depicted in figure 2.

Figure 1

Research network (left) and neighborhood of that same network (right). Dark node = a scientist; gray nodes = neighborhood.

Figure 2

Connected network (left) and unconnected network (right).

At each stage of inquiry, each scientist may choose to perform one of finitely many actions, such as conducting an experiment, running a simulation, making numerical calculations, and so on. We assume that the set of actions is constant through inquiry, and each action results (probabilistically) in an outcome. An outcome might represent the recording of data or the observation of a novel phenomenon. Scientific outcomes can usually be regarded as better or worse than alternatives. For example, an experiment might simply fail without providing any meaningful results. An experiment might succeed but give an outcome that is only slightly useful for understanding the world. Or it might yield an important result. In order to capture the “epistemic utility” of different scientific outcomes, we will represent them as real numbers, with higher numbers representing better scientific outcomes.Footnote ⁹

There is a set of possible states of the world, and the probabilities of outcomes (after an action) can depend on the state of the world. This captures the idea that many outcomes (e.g., experimental observations) depend on the world being a certain way. For example, the outcome of the detection of the bending of light during an eclipse depends on the state of the world—specifically, whether general relativity is true or false. Of course, detection is only possible if one sets up particular measuring apparatuses and so forth; some action must be taken by the scientist. We assume the probability of some outcome given an action and world is constant through time. Such an assumption may not be realistic. For example, a discovery is an outcome that can only happen once, regardless of how many times the scientist takes the relevant action.

Throughout, we will consider a running example of theory choice in cognitive psychology. Specifically, consider the problem of determining the nature of human concepts. There is a range of theories of concepts, including exemplar-based theory (Medin and Schaffer Reference Medin and Schaffer1978; Nosofsky Reference Nosofsky1984), prototype-based theory (Smith and Minda Reference Smith and Minda1998; Minda and Smith Reference Minda and Smith2001), causal-model theory (Rehder Reference Rehder2003a, Reference Rehder2003b), theory theory (Carey Reference Carey1985; Gopnik and Meltzoff Reference Gopnik and Meltzoff1997), and others. The states of the world in this example correspond to the actual cognitive representations that people have. Actions in the model correspond to a scientist acting to verify a particular theory: conducting experiments, performing mathematical derivations, and so forth.Footnote ¹⁰ Note that a scientist’s action in this model is not some psychological act or state but rather is an observable action that leads to an observable outcome. (We return to this point below.) Outcomes in the model are the importance, quality, and nature of the resulting research products. Clearly, outcomes depend on both what cognitive representations are actually in people’s heads (i.e., what our concepts really are) and also the type of investigation conducted. At the same time, outcomes are not a deterministic function of cognitive representations and the type of investigation, as many other factors can influence the research products (e.g., experimental design, measurement error).

Returning to the abstract model of scientific inquiry, we say that a learning problem is a quadruple , where Ω is a set of states of the world, A is a set of actions, O is a set of outcomes, and p is a measure specifying the probability of obtaining a particular outcome given an action and state of the world. In general, at any point of inquiry, every scientist has observed the finite sequence of actions and outcomes performed by herself, as well as the actions and outcomes of her neighbors. We call such a finite sequence of actions and their resulting outcomes a history. A method (also called a strategy) m for an individual scientist is a function that specifies, for any individual history, a probability distribution over possible actions for the next stage. In other words, a method specifies probabilities over the scientist’s actions given what she knows about her own and her neighbors’ past actions and outcomes. Of course, an agent may act deterministically, simply by placing unit probability on a single action . A strategic network is a pair consisting of a network G and a sequence specifying the strategy employed by each learner, , in the network.

To return to our running example, a particular psychologist directly knows about the research products of only some of the other psychologists working on the nature of concepts; those individuals will be her graphical neighbors. The experiments and theories considered and tested in the past by the psychologist and her neighbors are known to the psychologist, and she can use this information in her learning method or strategy to determine (probabilistically) which cognitive theory to pursue next. Importantly, we do not constrain the possible strategies in any significant way. The psychologist is permitted, for example, to use a strategy that says “do the same action (i.e., attempt to verify the same psychological theory) for all time” or to have a bias against changing theories (since such a change can incur significant intellectual and material costs).

Although we will continue to focus on this running example, it is important to note that the general model can be realized in many other ways. Here are two further examples of how the model can be interpreted:

Because each outcome is assigned an epistemic utility, we can speak of a particular action as being optimal in a given state of the world. An action is optimal in a state if its expected utility in that state is at least as high as that of any other action. For example, it might be that exemplar-based theories are the best theories for understanding concepts, and so the action of attempting to verify an exemplar-based theory will be optimal.Footnote ¹¹ Focusing on other theories might yield useful results occasionally, but on average they would (in this world) be worse.

Some learning problems are easier to solve than others; for example, if one action always dominates all others regardless of the state of the world, then there is relatively little for scientists to learn. This is rarely the case in science since the optimal actions typically depend on the state of the world. We are principally interested in such difficult problems. More precisely, recall that outcomes can be interpreted as utilities. Thus, for any state of the world ω, there is an expected value for each action that is constant throughout time. Hence, in any state of the world ω, there is some collection of optimal actions that maximize expected utility. Say a learning problem poses the problem of induction if no finite history reveals that a given action is optimal with certainty. In other words, the problem of induction (in this context) is that, for any finite history, there are at least two states of the world in which such a history is possible, but the sets of optimal actions in those two states are disjoint. Say a learning problem is difficult if it poses the problem of induction, and for every state of the world and every action, the probability that the action yields no payoff on a given round lies strictly between zero and one. That is, no action is guaranteed to succeed or fail, and no history determines an optimal action with certainty. For the remainder of the article, we assume all learning problems pose the problem of induction and will note whether a problem is also assumed to be difficult.

2. Individual versus Group Rationality

Undoubtedly, one goal of scientific inquiry is to eventually find the “best” theories in a given domain. In our model, “eventually finding the best theories” corresponds to performing optimal actions with probability approaching one in the infinite limit—this property is generally known as statistical consistency. In this section, we investigate several versions of the Independence Thesis that assert that individuals employing statistically consistent methods might not form consistent groups and that consistent groups need not contain consistent individuals. We find that, depending on exactly how the notion of consistency is construed, individual and group epistemic quality may either coincide or diverge.

Two questions are immediate. Why do we focus on statistical consistency exclusively, rather than considering other methodological virtues? And why is consistency characterized in terms of performing optimal actions, rather than holding true beliefs about the state of the world (and, hence, of which actions are in fact optimal)?

In response to the first question, we argue that statistical consistency is often the closest approximation to reliability in empirical inquiry, and therefore, employing a consistent method is a necessary condition for attaining scientific justification and knowledge. In the context of science, reliabilism is the thesis that an inductive method confers a scientist with justification for believing a theory if and only if the method, in general, tends to promote true beliefs. Unfortunately, even in the simplest statistical problems, there are no inductive methods that are guaranteed (with high probability) to (i) promote true rather than “approximately true” beliefs and (ii) promote even approximately true beliefs given limited amounts of data.

Suppose, for example, you are given a coin, which may be fair or unfair, and you are required to determine the coin’s bias, that is, the frequency with which the coin lands heads. Suppose you flip the coin times and observe heads. An intuitive method, called the “straight rule” by Reichenbach, is to conjecture that the probability that the coin will land heads is . Suppose that, in fact, the coin is fair. With what probability, does the straight rule generate true beliefs? If you flipped the coin an odd number of times, it is impossible for to be 1/2, and so the probability that your belief is correct is exactly zero. But it is worse than that. As the number of flips increases, the probability that the straight rule conjectures exactly 1/2 approaches zero. This argument does not show a failing with the straight rule: it applies equally to any method for determining the coin’s bias.

Therefore, reliability, in even the simplest settings, cannot be construed as “tending to promote true beliefs,” if “tending” is understood to mean “with high probability.” Rather, the best our inductive methods can guarantee is that our estimates approach the truth with increasing probability as evidence accumulates. But this is just what it means for a method to be statistically consistent.

However, one still might be worried that we have construed statistical consistency in terms of performing optimal actions rather than possessing true beliefs. Our focus is thus different from the (arguably) standard one in philosophy of science: we consider convergence of action, not convergence of belief. We are not attempting to model the psychological states of scientists but rather focusing on changes in their patterns of activity.Footnote ¹² This is why our running example from cognitive psychology focuses on the actions that a scientist can take to verify a particular theory, rather than the scientist accepting (or believing or entertaining or so on) that theory. Of course, to the extent that scientists are careful, thoughtful, honest, and so forth, their actions should track their beliefs in important ways. But our analysis makes no assumptions about those internal psychological states or traits, nor do we attempt to model them.

Moreover, we think that there are compelling reasons to focus on convergence in observable actions rather than convergence in a scientist’s beliefs. Suppose only finitely many actions are available and that a scientist cycles through each possible action in succession , and so on. As inquiry progresses, such a scientist will acquire increasingly accurate estimates of the value of every action (by the law of large numbers). Thus, the scientist will also know which actions are optimal, even though she chooses suboptimal as often as she choose optimal ones. So the scientist would converge in belief without that convergence ever leading to appropriate actions.

Consider for a moment how strange a scientific community that follows such a strategy would appear. Each researcher would continue to test all potential competing hypotheses in a given domain, regardless of their apparent empirical support. Physicists would occasionally perform experiments that relied on assumptions from Aristotelian physics, astronomers would continue developing geocentric models, geologists would use theories that place the age of the earth at 8,000 years, and so on. Such a practice would continue indefinitely, despite the increasing evidence of the inadequacy of such theories. We believe that commitment to a scientific theory (whether belief, acceptance, or whatever) ought to be reflected in action and, thus, that good scientific practice requires a convergence in not only the beliefs of the scientists but also the experimentation.Footnote ¹³

More radically, one might even question what role belief would have for individuals who continue to rely on increasingly implausible theories. What exactly does it mean for a scientist to believe in a theory that she uses as often as its competitors? We do not wish to delve too deeply into the issue of the relationship between belief and action, except to suggest that our would-be detractors would have to devise a notion of acceptance that radically divorces it from action.

With this background in mind, we now consider one formulation of the Independence Thesis: Do optimal methods for an isolated researcher prove beneficial if employed by a community of scientists who share their findings? More precisely, suppose we have a fixed learning problem and consider the isolated strategic network consisting of exactly one scientist. We say that the scientist’s method is convergent in isolation (IC) if, regardless of the state of the world, her chance (as determined by her method) of performing an optimal action (relative to the actual state of the world) approaches one as inquiry progresses.

In our running example, our concept-focused psychologist would be called IC if she, when working alone, would converge to always testing (and presumably publicly advocating) the optimal theory of concepts. For example, suppose she selects a focal theory at time by the following method: the theory that has yielded the best outcomes in the past is tested with probability , and a random theory is selected with probability . The psychologist’s method is an example of what are called decreasing epsilon greedy methods, and this particular decreasing epsilon greedy strategy is IC; the psychologist will provably converge toward testing the optimal theory with probability one. Interestingly, the method of always testing the best-up-to-now theory is not IC: testing the best-up-to-now theory can lead to abandoning potentially better theories early on, simply because of random chance successes (or failures) by suboptimal (or optimal) theories (Zollman Reference Zollman2007, Reference Zollman2010). The previous method is IC, precisely because our imaginary psychologist attempts to verify other theories sufficiently often to ensure that the best theory is not prematurely abandoned, while still acting to verify the best-up-to-now theory with increasing probability.

On first glance, convergence in isolation seems to be a minimal/necessary requirement for being a “good” method for learning. However, IC methods may have undesirable consequences when employed by a group of researchers. To see why, say that a collection of methods is convergent in an isolated group (GIC) if whenever every scientist in a research network uses a method in and every method in is used by at least one scientist, then each scientist’s chance of performing an optimal action approaches one as inquiry progresses. That is, a GIC collection of methods is one in which every method finds at least one optimal action (in the limit) whenever the other methods in the collection are present in the network. Surprisingly, as the following two theorems state, there is no necessary relationship between IC and GIC learners.

One example is a variation on the decreasing epsilon greedy strategy described above.Footnote ¹⁴

One example is a group of “preferred action” learners (described below).

The decreasing epsilon greedy strategy is an IC method that can lead to disastrous results when employed in groups. The psychologist’s method is IC because her probability of experimentation tapers to zero at the “right” speed. If that probability tapers too quickly (e.g., if the probability of randomly choosing a theory were ), then the method may fail to experiment frequently enough to find the best theory. In particular, if the psychologist’s rate of experimentation depends on the number of neighbors she has (e.g., a random sample rate of , where k is the number of neighbors), then she will converge to testing an optimal theory in isolation but will not necessarily converge to an optimal theory in a group. Essentially, the potential problem here is one of groupthink: by herself, each psychologist patiently figures out the right theory; in a group, however, she might converge on a suboptimal theory too quickly and get stuck.Footnote ¹⁵

As an example of the second theorem, consider the following collection of methods: for each action a, the method chooses a with probability (where n is the stage of inquiry) and otherwise does a best-up-to-now action. One can think of a as the “preferred action” of the method , although is capable of learning. For our psychologist, a is the “favorite theory” of concepts. Any particular method is not IC since it can easily be trapped in a suboptimal action (e.g., if the first instance of a is successful). The core problem with the methods is that they do not experiment: they either do their favorite action or a best-up-to-now one. However, the set is GIC, as when all such methods are employed in the network, at least one scientist favors an optimal action and makes sure it gets a “fair hearing.” The neighbors of this scientist gradually learn that is optimal, and so each neighbor plays either or some other optimal action with greater frequency as inquiry progresses. Then the neighbors of the neighbors learn that is optimal, and so on, so that knowledge of at least one optimal action propagates through the entire research network. In our running example, the community as a whole can learn, as long as each theory of concepts has at least one dedicated proponent, as the proponents of any given theory can learn from others’ research. Moreover, this arguably is an accurate description of much of scientific practice in cognitive psychology: individual scientists have a preferred theory that is their principal focus (and that they hope is correct), but they can eventually test and verify different theories if presented with compelling evidence in favor of them.

This latter example—and the associated theorem—supports a deep insight about scientific communities made by Popper and others: a diverse set of approaches to a scientific problem, combined with a healthy dose of rigidity in refusing to alter one’s approach unless the evidence to do so is strong, can benefit a scientific community even when such rigidity would prove counterproductive to a scientist working in isolation. The above argument also bolsters observations about the benefits of the “division of cognitive labor” that have been advocated by Kitcher, Strevens, Weisberg, and Muldoon, among others.

The definitions of IC and GIC focus on learners who have some measure of control over their research network. Although there are numerous examples and case studies of scientific communities trying to maintain control over their membership, most actual scientific communities are not so “insular.” We must also consider cases in which a learner or collection of learners is embedded in a larger community that could potentially contain individuals who have different epistemic goals, standards, or methods. Say a method is universally convergent (UC) if in any state of the world, the researcher chooses optimal actions with probability approaching one as inquiry progresses, regardless of the network in which the researcher is embedded. Similarly, say a set of methods M is group universally convergent (GUC) if for all networks such that every method in M is employed at least once and everyone employing methods in M is connected via “informational paths” consisting only of scientists also employing methods from M, then each researcher employing some method in M chooses optimal actions with probability approaching one as inquiry progresses.

UC and GUC methods are those that are, in a specific sense, resistant to the other individuals in the network. A UC method converges to an optimal action, for example, even when it is surrounded by methods that produce suboptimal actions for all eternity. Notice that, in all learning problems, every UC method is IC, and every GUC collection is GIC. Why? Isolation is simply a special type of network structure, and since UC methods and GUC collections converge regardless of network structure, they must converge for these special cases. As one might suspect, UC and GUC are strictly stronger epistemic standards than IC or GIC.

Perhaps the most commonly studied strategies witnessing the above theorem are called reinforcement learning (RL) strategies. Reinforcement learners begin with an initial, positive, real-valued weight for each action. On the first stage of inquiry, the agent chooses an action in proportion to the weights. For example, if there are two actions and with weights 3 and 5, respectively, then the agent chooses action with probability and with probability . At subsequent stages, the agent then adds the observed outcome for all the actions taken in his neighborhood to the respective weights for the different actions. RL has been used as a descriptive model of learning opponents’ behavior in games and various cognitive learning processes.Footnote ¹⁶ It might also be recommended as a normative model for induction, as it is consistent when employed in isolation (Beggs Reference Beggs2005). RL strategies are IC but not UC, and groups of such learners can be GIC but not GUC.

Interestingly, however, the disconnect between group and individual epistemic norms found in isolation does not carry over in the same way to universal convergence, as shown by the following two theorems. In particular, there is a connection, but only in one direction.

One example is a set of “preferred action” methods (described above), which is a GUC collection of not-UC (not-IC) methods.

The relationships between IC, GIC, UC, and GUC can thus be neatly summarized by figure 3. The above five theorems provide qualified support for certain formulations of the Independence Thesis, as they show that, when epistemic quality is understood in terms of statistical consistency, there are circumstances under which individual and group rationality diverge.

Figure 3

Of course, one might object that our arguments rely on one particular formal model of inquiry that makes several unrealistic idealizations: network structure is constant, informational access is symmetric, and so on. One might continue that we have considered only stringent criteria of success and that there are any number of other standards by which one might evaluate epistemic performance of individuals and groups. Such an objector might conclude that the above theorems say little about the relationship between individual and social epistemology in the “real world.”

First, note that a similar divergence between individual and group epistemic performance (measured in terms of statistical consistency) must emerge in any model of inquiry that is as complex as our model (or more so). For any model of inquiry capable of representing the complexities captured by our model, we would be able to define all of the methods considered above (perhaps in different terms), define appropriate notions of consistency, and so on. Thus, each of the above theorems would have an appropriate “translation” or analog in any model of inquiry that is sufficiently complex. In this sense, the simplicity of our model is a virtue, not a vice.

Moreover, we agree that our model is not the single “correct” representation of scientific communities. In fact, we would argue that no formal model of inquiry is superior to all others in every respect. Clearly, there are other criteria of epistemic quality that ought be investigated, other models of particular scientific learning in communities, and so forth. However, significant scientific learning problems can be accurately captured and modeled in this framework, and the general moral of the framework should hold true in a range of settings, namely, that successful science depends on using methods that are sensitive to evidence and can learn, while avoiding community groupthink.

3. Conclusion

In this article, we have investigated several versions of the Independence Thesis that depend on different underlying notions of individual rationality (IC and UC) and group rationality (GIC and GUC). We have found some qualified support for the Independence Thesis; IC is independent of GIC. However, considering a stronger notion of rationality, we found that UC is not independent of GUC, although the entailment goes in one direction only.

Our formulations of these different notions of rationality illustrate important distinctions that have been previously overlooked. For example, the different properties of IC and UC (GIC and GUC, respectively) methods illustrate that, when considering individual (or group) rationality, one must consider how robust a method (or set of methods) is to the presence of others employing different methods. Subtle differences in definitions of individual and group rationality, therefore, can color how one judges the Independence Thesis.

In addition to illustrating the importance of this distinction, we have provided some formal support to those philosophers who have pushed the Independence Thesis (primarily on the basis of historical evidence). It is our hope that this proof will demonstrate the thesis, in a mathematically rigorous way, and thus illustrate the limitations of discussions of scientific method that consider only the properties of inductive methods when employed in isolation, as is common in many formal models of scientific inquiry (e.g., Bayesianism, belief revision, formal learning theory). Finally, our arguments provide additional support to those who argue that the empirical study of science must include both the individual and the group properties of scientific groups if the prospects for scientific progress are to be understood properly.