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A dynamical systems approach to causation


Our approach aims at accounting for causal claims in terms of how the physical states of the underlying dynamical system evolve with time. Causal claims assert connections between two sets of physicals states—their truth depends on whether the two sets in question are genuinely connected by time evolution such that physical states from one set evolve with time into the states of the other set. We demonstrate the virtues of our approach by showing how it is able to account for typical causes, causally relevant factors, being ‘the’ cause, and cases of overdetermination and causation by absences.

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  1. Depending on whether causes and effects are characterised by specific token level or by general type level descriptions our framework is equally able to account for both token and type causation (Sect. 2.2).

  2. Counterfactual (Lewis 1986), and other difference making accounts (Woodward 2003) follow the first route, whereas physical accounts of causation (Fair 1979; Salmon 1984; Dowe 2000; Kistler 2006) are more concerned about the second.

  3. Some have already tried to approach causation from the perspective of dynamical systems theory. The view that is closest to our proposal is Christopher Hitchcock’s sketch of what he calls ‘Laplacian causation’ (Hitchcock 2012, pp. 46–51). Holly Andersen (2017) has recently published an account of causation that relied on dynamical systems and phase-space terminology. However, her aim was to develop an information-theoretic approach, thus the framework she offers is fundamentally different from ours. In a recent series of papers List and Pivato (2015) utilised a dynamical systems approach and applied it to a range of philosophical phenomena—but not to causation. Earlier List and Menzies (2009) analysed higher-level causation—but not in the framework of dynamical systems approach. Our paper unites the two programmatic aims: it combines higher-level causation with lower-level dynamical processes.

  4. Hence, the scope of our approach is limited to deterministic systems. This limitation does not necessarily exclude the application of our analysis to quantum mechanical systems, as there exist viable no-collapse interpretations (such as the Bohm–de Broglie theory) that render ordinary quantum mechanics deterministic in the relevant sense.

  5. Of course, the number of relevant negative truths (conditions that do not hold in a particular situation) is always infinite. A descriptive state, however, is determined by only those properties that a particular causal discourse actually invokes. Introducing new properties into the characterisation of causes and effects results in a more fine-grained partitioning. Type-level descriptions provide coarser-grained partitioning, token-level descriptions provide finer-grained partitioning (since token-level descriptions are still macro descriptions—the partitioning they impose upon the state space still consists of regions).

  6. Though developed independently, this aspect of our account resembles what Hitchcock calls ‘region cause’ (Hitchcock 2012, p. 49).

  7. Our everyday causal parlance is very sensitive to the temporal aspects of causal relations: it is not enough for a given state to evolve into another state within some finite time, it has to evolve into that within a reasonable time bound that is characteristic of the typical length of the causal influences that a particular causal discourse tries to describe. Individual physical states of a region might need slightly different amounts of time to evolve into another region, but these fall within an interval that corresponds to the characteristic time bound. For an illustration see the caption to Fig. 11.

  8. For a discussion of this issue in less abstract terms see Sect. 4.1.

  9. This notion of robustness captures the traditional understanding that robust causal relationships hold in a variety of backgrounds, as regions corresponding to different backgrounds intersect with the cause region cutting sub-regions out of the original cause region. If a causal relation doesn’t hold in a given background, then the majority of the physical states from the corresponding sub-region will not evolve into the effect region.

  10. All these considerations, then, mutually constrain the sizes of the cause and effect regions, i.e. the level of detail in which causes and effects can be specified. Due to limitations in space, here we can only propose that this is what ultimately grounds intuitions about the proportionality of causation (Yablo 1992).

  11. We say that a descriptive state is on the backwards trajectory of a physical state iff it is crossed by the trajectory that evolves into said physical state.

  12. Two further issues need to be emphasised here. First, the partitioning offered by a causal discourse is typically not specific enough for the entire time evolution of the trajectory in question, and thus it won’t be able to carve regions out from the state space along the entire trajectory such that from these the majority of the points evolve into the effect state. Therefore, descriptive states that a trajectory—followed backwards—crosses are not always projective states, and thus for any given effect state there will probably be a corresponding principle projective state. Second, intuitively one may characterise the principal projective state as the first projective state into which the forward trajectory (that ends up in the effect state) enters after leaving a descriptive state that is not yet a projective state. Taking into account our restriction on characteristic time development this is another useful way to characterise our notion of principal projective state; however, strictly speaking, there is no guarantee that the principal projective state is unique. If one followed the trajectory further backwards it may be the case that after passing through several non-projective states the trajectory again enters into another (principal) projective state. Indeed when the dynamics is ergodic one would expect that the backwards trajectory would pass through again and again the same principal projective state. See also our discussion of the typical principal projective state in Sect. 4.2.

  13. In the literature, the notion of relevance is also used to distinguish between different descriptions of the same phenomenon (relying on different properties that are in a supervenience relation). In such cases, the different projective states cover the same section of a trajectory in different ways, but the trajectory enters and leaves these states at the same time (as supervenience is a synchronic relation). A full discussion of how our approach could be used in those cases requires a standalone paper. (See also the Conclusion).

  14. In addition to such actuality consideration, causal selection can also be driven by normative (see Fn. 15) and typicality considerations (Hart and Honore 1985). Due to limitations in space we can only indicate that besides typicality (see Sect. 3.1), the notions of atypicality and abnormality, which play a central role in selecting ‘the’ cause of an effect in typicality-driven cases, can also be accounted for within our approach. A full treatment of this issue shall follow in a subsequent paper.

  15. McGrath (2005) argues that causal intuitions in omission cases are governed by normative expectations. Normative expectations are grounded either in one's knowledge about what happens typically under some circumstances, or in morally or socially dictated obligations. The latter cases, however, fall outside what any theory of causation should rationally be expected to account for.

  16. Even though more thorough investigation is required to assess the compatibility of our approach with indeterministic dynamics.


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The authors wish to thank Jonas Christensen, Matteo Colombo, Markus Eronen, Robin Hendry, Andreas Hüttemann, Beate Krickel, Mark Pexton, Stathis Psillos, Miklós Rédei, two anonymous referees for this journal, and all the members of the audience at the workshops and conferences in Aarhus, Budapest, Cambridge, Durham, Düsseldorf, Groningen, Krakow, Lille and London for their helpful comments on earlier versions of this paper.


This work has been supported by the FWO Postdoctoral Fellowship 1.2.B39.14 N and the DFF-EU MCA-COFUND Mobilex Grant 1321-00165 (PF); the National Research, Development and Innovation Office K-115593 (BGy and GH-Sz); the Hungarian Scientific Research Fund OTKA K-100715 (GH-Sz); and the Durham Emergence Project and the MTA BTK ‘Lendulet’ Morals and Science Research Group (GK).

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Fazekas, P., Gyenis, B., Hofer-Szabó, G. et al. A dynamical systems approach to causation. Synthese 198, 6065–6087 (2021).

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  • Causation
  • Physical causation, folk causation, dynamical systems
  • State space
  • Time evolution