Office for Applied Complexity

The Demons’ Butterfly Collection

Text and illustrations by Joshua Johnson

As a manner of illustration, Roman Frigg introduced us to three demons: Laplace’s Demon and his two apprentices, the Senior apprentice and Freshman apprentice 1.

In the early 19th Century Pierre-Simon Laplace proposed a thought experiment regarding causality, determinism, and classical physics. His notion was that if the position and momentum of all the atoms of a system were known, then all past or future states of that system could be predicted from this information. His demon is a fictional embodiment of this idea:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

—Pierre Simon Laplace, A Philosophical Essay on Probabilities 2

Frigg redefines Laplace’s Demon to have three capabilities for modelling a system:

1. Computational Omniscience – the demon has perfect computational power, and is able to calculate all values of the system arbitrarily fast.
2. Dynamical Omniscience – the demon has perfect knowledge of the true time evolution of the system.
3. Observational Omniscience – the demon has perfect knowledge of the true initial conditions of the system.

In our own attempts at modelling complex systems, we lack for all three of these conditions, but the point of this continued thought experiment is to understand to what degree mere mortals might be able to approach a perfect system model. Roman therefore introduced us to two other characters, the Senior apprentice and the Freshman apprentice. All three demons are granted computational omniscience, however, only Laplace’s demon has dynamical and observational omniscience. The Senior apprentice is given the gift of dynamical omniscience, but lacks observational omniscience. The Freshman apprentice has only computational omniscience. To put it another way, the Senior apprentice has the perfect model of the system, but does not know the initial conditions of the system, while the Freshman apprentice has neither.

Frigg related observational omniscience to the problem of the Butterfly effect. As the popular illustration has it, a butterfly flaps its wings and this small perturbation in the wind causes a hurricane half way across the world. The idea, drawn from Chaos Theory, has been popularized in  culture, and has appeared in such films as Jurassic Park (1993) and, of course, The Butterfly Effect (2004). In Jurassic Park, for instance, small changes to the dinosaur’s reconstructed DNA introduces a capacity to reproduce, which the scientists thought they had prevented, causing unexpected chaos as the dinosaurs escape the control of their handlers.  Whatever the relative scientific or artistic (de)merits of these films, the basic notion is that small changes in a system’s initial condition conditions can have a significant effect on the system at a larger scale, making prediction of the system’s behavior difficult.

Laplace’s Demon’s true system model.

Neither the Senior or the Freshman apprentice have access to the initial conditions of the system, and so must use clever techniques to work around this liability if they are to prove their ability and make useful (decision relevant) predictions on the evolution of the system.

The Butterfly effect means that even minute differences in the initial conditions will affect the trajectory of the predictions, so both apprentices apply probability distributions in place of the initial values. This allows them to posit predictions which, while not exact to the true system evolution, fall within an effective range of the system’s likely evolution. As Frigg notes, this works very well for the Senior apprentice, whose predictions fit the general evolution of the system, but the Freshman apprentice has a problem.

The Senior apprentice’s probability model is illustrated by the color distribution in the first image, and the Freshman apprentice’s model by the color distribution in the second. The true system is depicted in white. 

Since the Freshman apprentice does not have the perfect model, the demon must approximate a model, as well as the initial system conditions. The application of a probabilistic distribution allows both the Freshman and Senior demons produce models that “smear” around the expected development of the system, but because the Freshman apprentice lacks a proper model, the “smearing” evolves along an entirely different trajectory than the actual system’s evolution. Even if the Freshman cheats, and steals the true initial conditions from Laplace’s demon, the errors inherent the model will cause the predictions to wildly deviate from the actual trajectories of the system.

Frigg identifies this problem as a Structural Modelling Error (SME) that cannot be overcome in the same way that probabilistic modelling compensates for the butterfly effect. Frigg and his colleagues have dubbed this problem the “hawkmoth effect” because of its analogous position to the butterfly effect in predictive modelling. Like the butterfly effect, where small perturbations in initial conditions can have unexpected consequences for the system’s evolution, small errors in the structure of the model may have unpredictable consequences for the accuracy of any projections.

The hawkmoth effect poses enormous challenges for the predictive success and decisional factors that must be accounted for when choosing our models and how we approach our representations of the world. For more on these challenges and additional conversational points check out our audio session with Roman Frigg:

Audio from our session at Tate Britain. In addition to Roman Frigg, participants include the anthropologist Benedetta Rossi and OfAC members Diann Bauer, Victoria Ivanova, Joshua Johnson, Suhail Malik, Patricia Reed, and Natalia Zuluaga. Special thanks to Tate Britain for hosting and supporting this event.