Why Arguments are (almost) Never Convincing: A Dynamical Systems Approach to Belief Change

Introduction

What I want to do in this piece is outline a perspective for thinking about belief systems and how people change (or don’t change) their beliefs in response to new arguments and evidence. The key observation that motivates this analysis is that in general, when people have a particular entrenched perspective with respect to an issue or subject, it is very rare that they find any new evidence or arguments sufficiently persuasive to significantly change their beliefs. I have been thinking about a model that would have to explain why this is the case, a model which incorporates concepts from physics and dynamical systems theory. That might sound very complicated, but I think the key idea is relatively simple. I am not claiming that this approach is exhaustive or completely accurate, but rather that it may be a useful way of thinking about when and why people change their beliefs, and why they seldom do. My focus here will be on disputes surrounding complicated and controversial matters such as politics, religion, and philosophy, though the model my be applicable elsewhere as well.

Background

Imagine a bowl with a marble in the middle, lying stationary at the bottom of the curve of the bowl. If we jiggle the bowl around, or push the marble up one side or another, it will roll back down towards the centre. It may jiggle around for a while, rolling up one side and down the other, but eventually it will return to rest at the centre of the bowl. This behaviour corresponds to that of a potential energy well in physics, whereby a system has a state in which its energy is lowest, to which the system tends towards as a result of the overarching tendency to reach its lowest energy state. Perturbations away from this minimal energy state will generally only be temporary, and eventually the system will return towards its ‘preferred’ state. In the language of dynamical systems, this state is described as a stable equilibrium, because if the system (in our example the system consists of the bowl and the marble) is perturbed slightly one way or the other, it will eventually return to its initial resting equilibrium state.

Now imagine that we placed two bowls next to each other, and joined together their edges so that they were connected by a smooth, curved edge, sort of like two sinks nested next to each other in the same bench. If we placed our marble exactly halfway in between the two sinks, we could get it to rest there without moving. However this equilibrium state, unlike the one where the marble is in the middle of one of the bowls, is unstable, since a small nudge in either direction will send the marble rolling into one of the bowls, never to return. This illustrates the key point that in contrast to stable equilibria, unstable equilibria are not robust to small perturbations.

Now imagine that we place a third, much smaller and shallower bowl in between our larger bowls (again with the edges smoothly joined), but placed on a platform so that its top is level to the top of the other bowls. This may be slightly more difficult to imagine, but essentially it would correspond to a shallow sink placed in the same bench in between two deeper sinks. A marble placed in the centre of this smaller will remain there and will return when subject to small shocks. However if we push the marble with enough force, it will have sufficient energy to exit the central bowl, travel over the curve connecting it to one of the larger bowls, and fall down to the centre of this bowl. From this location, it will obviously not be able to return to its original position in the shallower, central bowl. In the language of dynamical systems theory, this central bowl is called a locally stable equilibrium – it is robust to small perturbations, but not to larger ones. Note that it is also possible in theory to knock the marble out of the larger bowl all the way over the lip and back into the shallower central bowl, however this would take a very large push indeed. Thus we say that the larger bowl is a more stable, ‘lower energy state’ (in physics terminology) than the central bowl.

A final concept that I need to introduce is that of a dynamical system. The precise technical definition of a dynamical system is not of interest to me here, and would detract from the key logic of the argument. What I mean by ‘dynamical system’ is in particular a system which changes over time in a manner which is (in some sense) ‘recursive’, such that changes of the system depend upon the current state of the system. A simple example would be differential equations, which are equations whereby the value of one variable (say x) depends on the rate at which that variable is changing with time (dx/dt), which itself depends upon the current value of x. The key property is that many such dynamical systems can evolve in quite complicated ways, leading to some solutions which are stable (corresponding to equilbria discussed above), and others that are not. Dynamical systems evolve over time in what is called the state space, which corresponds to the set of possible values that all the variables could take. A simple example of a dynamical system is a pendulum. The system is dynamic because the velocity of the pendulum depends on the height of the pendulum, which in turn obviously depends on past velocity values, producing a potentially complicated temporal trajectory. The state space consists of the possible values of the height of the pendulum and the rate at which that height is changing. As the pendulum moves from side to side, speeding up and slowing down under the force of gravity, the pendulum moves through the state space, constantly changing its velocity and position values.

The Model

Having outlined some key concepts, I will now apply these ideas in understanding belief formation and change. The key idea is to consider the process of belief formation as a dynamical system seeking to find the ‘lowest energy’ state. Imagine viewing our set of bowls from above. Our marble corresponds to a particular person, and the marbles position in and around the bowls represents that person’s current set of opinions and beliefs about a specific subject; ‘where they are at’ intellectually. We can describe movement in three dimensions: north and south (the ‘y axis’), east and west (the ‘x axis’), and up and down (which corresponds to the depth below the top of the bowl). The position along the x-axis represents one’s opinion on one particular specific question, while the position on the y-axis represents one’s opinion on a different particular question. The depth below the top of the bowl represents one’s degree of confidence in one’s overall set of positions. It should be noted that for any sufficiently complicated issue there will be far more than two particular questions of relevance – they may be dozens or even hundreds. Mathematically there is no limit to how many dimensions a dynamical system can have, however for simplicity of visualisation we will stick with only two for this example, always bearing in mind that for real world examples we would always wish to extrapolate out the analysis to many more dimensions.

The system is said to be dynamical because each individual evaluates the x- and y-axis positions interdependently. That is, it is not the case that they arrive at a position on the issue corresponding to the y-axis and then independently decide upon the issue corresponding to the x-axis. Rather, they consider both issues simultaneously, so that the plausibility of a particular position along x is judged in relation to the position along y, which in turn is judged with respect to the position along x, and so on. The overall degree of confidence (depth) then depends upon how well one’s views on the two issues cohere or fit together, and so will also vary in accordance with the positions along the x- and y-axes.

Sometimes it may seem to us that with respect to a particular issue, different people have opinions that are spread ‘all over the map’, with each person being similarly confident in their individual set of beliefs. In the context of our model, this would correspond to a situation where hundreds of marbles were thrown into a flat-bottomed swimming pool, each at the same depth (degree of confidence), spanning the entire range of views along the x- and y-axes. In practise, however, I think this is a relatively rare outcome. More typically there are a few particularly deep wells that seem to serve as attractors for opinions, with only a few people residing outside of these deeper wells. Each of these wells, or deep bowls to use our previous language, corresponds to a particularly common set of positions on the subject in question. The reason these wells are so common is because they are self-sustaining, or in the language of dynamical systems, they are stable equilibria. Small changes in beliefs along either the x- or y-axes will not have any significant long-term effect on the system (the individual’s set of beliefs), which eventually will return to its initial state at the bottom of the well. The reason few people reside in between the major wells is because these positions, being much ‘higher up’ (corresponding to the connections between bowls discussed above) are unstable equilibria, where small perturbations in beliefs will lead to that individual ‘rolling down’ into one or other of the surrounding wells, arriving at a new stable equilibrium.

Applying the Model

To provide an example for this rather abstract model, consider the issue of the truth of Christianity. In this broad issue, two (among many other) specific questions would be that of whether the cosmological argument for the existence of God is found to be persuasive, and whether the historical evidence for the resurrection is found to be compelling. In theory, any possible combination of positions on these two issues is possible. In practise, however, probably only three main subsets of beliefs will be found: those who find neither argument very compelling (atheists and agnostics), those who found both compelling (Christians), and those who find only the cosmological argument compelling (some Muslims, Jews, and generic theists). Of course other combinations and intermediate positions are possible, but in general views will tend to cluster around these three positions. The reason for this, I think, is that these positions constitute attractor ‘wells’, such that people whose views are nudged in the direction of one of the wells are likely to fall into that well, seeking the lowest ‘energy state’ (i.e. a position with a high self-sustaining degree of confidence).

I think there are two processes key at work that lead to this outcome. The first is the interdependent way in which people analyse different specific arguments: those who are compelled by the cosmological argument are likely to find the evidence for the resurrection more persuasive, which in turn can feed back and increase one’s confidence in the cosmological argument. Conversely, a skeptical attitude towards one of these is likely to contribute to a skeptical attitude towards the other, thereby in turn reinforcing the original skeptical belief. In this way particular clusters of beliefs corresponding to ‘potential wells’ are likely to be far more stable than other possible clusters of beliefs, and thus result in these clusters being far more populated. The second process is that people tend to seek greater confidence and certainty, and this is likely to be found when their set of opinions on particular issues is mutually coherent and reinforcing. Again, this leads to certain particular clusters of beliefs, corresponding to the self-sustaining potential wells, to be more highly populated than other possible positions.

The combined effect of these two processes explains why people with intermediate or conflicting views on many particular questions are relatively rare. These people are not highly confident because their views are not mutually reinforcing. As such they seek out new arguments and evidence and are much more likely to change their views in the direction of greater coherence. Intermediate positions are thus unstable or only locally stable, so small perturbations (consisting of exposure to new arguments and evidence) are much more likely to push them into more stable potential wells. Once in one of these wells, however, opinions are much more stable. Even when confronted with potentially powerful counter-evidence on one particular question, the combined force of all one’s other positions (forming the coherent, mutually-reinforcing position) serves to pull one back to the original, stable position near the bottom of the well.

The only time when most people will move out of their wells is when they are subject to very large shocks, or enough moderate shocks in a relatively short span of time. Large enough shocks, or enough additive smaller shocks, may be enough to push someone out of their potential well and into the unstable area that lies between opposing wells. From there they may eventually return to their original well, or find themselves in an opposing well. Either way, it is unlikely that they will remain in the intermediate position for long, since this corresponds to an unstable or only locally stable equilibrium, where beliefs are not mutually reinforcing to a large degree and hence overall levels of confidence (corresponding to the depth of the potential well) remain low.

Virtues of the Model

This model can allow us to understand not only why people tend to cluster around a few particular positions (sets of beliefs about particular questions), and why people seldom change their belief when exposed to new evidence, but also why people sitting in opposing ‘wells’ (stable sets of beliefs) tend to react in exasperation at the ‘irrationality’ of each other. Consider the example of an atheist providing one argument in favour of their position. A christian evaluates the argument not in the context of the atheist’s set of beliefs (where the argument is persuasive), but from the context of their own set of beliefs. Because their set of beliefs is very different, and also because it is mutually coherent and stabilizing, the christian will either not consider the argument to support atheism at all, or they will not regard it as sufficient evidence to move from their current position (again, because their current position is a stable equilbrium, robust to even moderate shocks). The atheist seeing this intransigence to (from their perspective) such an obviously reasonable argument, regards the Christian as unreasonable and irrational. Exactly the same process occurs in reverse when the Christian presents arguments in favour of their viewpoint. As such both sides become polarised, viewing the other as unreasonable or irrational.

This model can also explain another puzzling phenomena: when the same evidence is claimed by different people as supporting their own, mutually incompatible positions. In the context of our model, this corresponds to a push in the same ‘direction’ leads to very different subsequent movements in the state space of possible positions. The explanation for this behaviour is that the way people respond to evidence and arguments (‘pushes’ or ‘perturbations’) in a dynamical system does not depend only on the size and direction of the push, but also on one’s current position in state space (i.e. one’s current set of beliefs). As such, the very same evidence (push in the same direction) can be interpreted by both the atheist and the christian as supporting their existing set of views. This renders the idea that ‘evidence speaks for itself’ as essentially impossible, since the manner in which evidence is interpreted depends upon one’s current set of beliefs.

Conclusion

I think it sheds quite a bit of light onto the process of belief formation and change, including explaining why people tend to congregate into groups with particular sets of beliefs, why once arriving at such a stable equilibrium in a ‘potential well’ people are unlikely to change their beliefs, how different people can react so differently to the same evidence, and why people on both sides of an issue can plausibly see each other as being intransigent and irrational. I think the model can also account for why substantial belief change is rare but possible, since it requires sufficiently large or sufficiently many shocks to one’s beliefs, and these shocks are (plausibly, in many cases) randomly distributed across people, substantial belief changes will occur but only relatively infrequently. Supposing we take this model as useful and informative (though certainly not complete), how should we respond? What effect, if any, should this have on our discourse and belief forming process? My honest answer is that I don’t really know, I’m still thinking this through. I think that overall the model paints a pessimistic picture of prolonged and resilient disagreement, where each side regards itself as rational by its own lights. I suspect more can be said here, but at the moment I’m still uncertain as to where to go with this analysis. Nevertheless, I think it does highlight the importance of intellectual humility and of respectfully considering opposing positions from a sympathetic viewpoint.

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