Can Causal Models Include Variables with Their Time-Derivatives? Naftali Weinberger Tilburg Center for Logic, Ethics and Philosophy of Science Time and Causality in the Sciences June 8th, 2017

Principle of the Common Cause (1) iPad Happiness (2) iPad Happiness Income (3) iPad Happiness Common Restriction: Does not (generally) apply to logically, mereologically, or conceptually related variables Discuss variables

Variable Distinctness Distinctness still assumed in contemporary approaches (Causal Markov Condition, Modularity) Central Question: Are variables and their time- derivatives distinct in the relevant respect? General project: How do we deal with variables that are ‘about’ other variables X and chance of X Dispositional properties and their effects X and is perceived as X

Overview Review a debate from the philosophy of physics on the causal role of instantaneous velocity Explain why similar issues arise in the graphical causal modeling framework and show how they can be resolved Draw conclusions about the semantics of derivatives in causal models

Instantaneous Velocity Textbook Definition: V0 Xt δx δt v0 v0 time

Is Instantaneous Velocity a Cause Reductionism: instantaneous velocity in physics is nothing more than the time- derivative of distance Question: Can the the reductionist account for the (alleged) causal role of velocity as the cause of the future trajectory of the object

Lange’s Argument a0 v-1 v0 v0 Xt If Reductionism, then instantaneous velocity at t would be partly constituted by events in the future, and V0 would not be a cause of all points in its future trajectory What if v0 were just the ‘derivative from below’? If velocity (and acceleration) were defined from below, then it could not also serve as an effect, and there could be no causal chains

Lange’s proposal V0 is not reducible to any fact about the object’s trajectory. If an object’s velocity is V=v at t0 and the object continues to exist with its trajectory undisrupted, then the derivative from above at t0 will equal v His proposal delivers the causal ordering: ForceAccelerationVX

Easwaran’s Response There is a coherent reductionist picture v0 Xt There is a coherent reductionist picture Acceleration at t=0 is the derivative from above of the derivative from below of displacement Acceleration is only an effect, but it is constituted by something (velocity) that can serve as a cause

To the Causal Models!

From Metaphysics to Methods What problems arise in trying to represent variables and their derivatives in causal models? Given a causal model in which XY, it is possible to change the value of Y via an intervention that sets the value of X These models use variables to represent discrete events

Interventions on X If V is defined at t, then interventions on X must also be interventions on V (if V is to remain well-defined) Xt Distinction between ‘from above’ and ‘from below’ implies discontinuity t time

Interventions on V Vt V’t Xt t Distinction between ‘from above’ and ‘from below’ implies discontinuity time An intervention on V changes the velocity from above. Now X is continuous, but not V.

Lessons One cannot intervene on a lower-order derivative if there is a well-defined higher- order derivative at that time Even though one cannot directly intervene on lower-order derivatives, they still play an indispensible role as inputs for the future trajectory of the object

Representation When we include a variable with its derivatives, we need new notation to represent their relationship Including a variable and its derivative does not lead to problems if: One cannot intervene on lower-order derivatives All effects of the highest-order derivative go via the lower-order derivatives A V X

Derivatives in Causal Models Existing modeling methods explicitly satisfy the proposed constraint Voortman, Dash and Druzdzel 2008

Why Use Derivatives in a Model? By including a variable with its time- derivative, we decompose the evolution of an entity into two components: The state of the object at a time based on the accumulation of prior changes The rate of change of the object which - combined with the object’s present state - determines its next one

Derivatives Without Continuity Use (the representational device developed for) derivatives for fine-grained time scales s.t.: Rate at which system is sampled >> rate at which we can intervene When we don’t use derivatives, we assume that: Rate at which system is sampled << rate at which we can intervene

Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the priority program "New Frameworks of Rationality" (SPP 1516). My thoughts on this topic have been influenced by ongoing discussions with the following people: Karen Zwier, Shannon Nolen, Jonathan Livengood and Adam Edwards

Conclusions Lange and Easwaran are correct that there are problems in treating velocity both as a cause and an effect Modeling solution: model the highest-order derivative as an effect and the rest as causes and introduce notation to denote the relevant constraints This modeling device can be applied to model the complex temporal dynamics even of non- continuous systems

Here’s What I’m Reading. Join Me! Iwasaki, Y., & Simon, H. A. (1994). Causality and model abstraction. Dash, D., & Druzdzel, M. (2001). Caveats for causal reasoning with equilibrium models. Hausman et al. (2013). Systems Without a Graphical Causal Representation. Voortman, M., Dash, D., & Druzdzel, M. J. (2012). Learning why things change: the difference-based causality learner. Lange, Marc. "How can instantaneous velocity fulfill its causal role?.” Easwaran, Kenny. "Why physics uses second derivatives.” Hoover, Kevin, (2015). The Ontological Status of Shocks and Trends in Economics” Ismael, J. (2016). How physics makes us free. Myrvold, Wayne. “Steps on the Way to Equilibrium” Norton, John. (2016). The Impossible Process: Thermodynamic Reversibility” Frisch, Matthias (2016). Causal Reasoning in Physics