This post catches up with a nice paper published last year by Sam Baron and Baptiste Le Bihan called “Causal Set Theory is (Strongly) Causal.” (A post about a previous, related paper by the same authors is here.) Causal Set Theory (CST) is an approach to quantum gravity that features discrete, causally linked foundational elements, from which the spacetime of general relativity (GR) is supposed to emerge as a large-scale approximation. In this paper, the authors are not assessing the prospects for success of the CST program, but asking a philosophical question: How exactly are CST’s links “causal”, and how does this relate to other notions of causation?
What follows are my notes on the paper. At the end, I briefly speculate about how CST might connect to my preferred approach to causation and ontology.
In the Introduction (section 1), the authors say:
Focusing on the standard sequential growth dynamics for CST, we argue that if the dynamics is treated as a real physical process, then CST makes use of causation of a specific sort, whereby causal set elements causally depend for their existence on previous elements. To show this, we use the framework of interventionism (Pearl, 2000; Woodward, 2005), which is a helpful tool for identifying causal relations of various types. Treating the growth process as a real physical process is controversial (see Arageorgis 2016; Huggett 2014; Wuthrich and Callender 2017). As we shall see, however, our analysis might help to reimagine the growth process by treating it as a causal process rather than a process of temporal passage or becoming. This is significant, since much of the resistance to treating the growth process as a real process stems from the conceptualisation of it as a process of becoming (p.2).
I have a few preliminary comments. The authors will use the interventionist framework as a tool for assessing the presence and nature of causal links. The relevant notion of causation here is one of dependence (sometimes called “difference-making”): causal dependencies are identified through evaluating a particular counterfactual scenario that imagines making a targeted change to a set of variables. For comparison, there is a distinct conception of causation—production—that connotes entities bringing about change in one another via a connection. Production is the causal notion embodied in mechanistic frameworks (it is also the conception fleshed out in my own work on causal process theory—see prior post or my recent paper).
Looking at this quote, it is also interesting that some critics of taking CST dynamics seriously think the “becoming” inherent in the growth process must imply a temporal form of passage. In contrast, I find it easy to conceive of becoming as rooted in causation, with temporal passage seen as deriving from this more fundamental basis.
Section 2. Clarification and Motivation
Here the authors first distinguish between mere causal connectibility (a necessary condition derived from the spacetime structure of relativity), which they call a “weak” form of causation, and a stronger notion called actual causation (p. 3). Next, they distinguish two kinds of actual causation:
Our interest is not just in any type of actual causation. Rather, we are interested in actual causation of a very specific variety. For, as we see it, there are two distinct types of actual causation.
Think again of the general relativistic case. One type of actual causation might hold between whatever is located at two spacetime points. So, for instance, we might look at how light emitted from point x affects a detector located in its future light-cone, at point y. Call this type of actual causation: material causation. Another, more radical type of actual causation might hold between the spacetime points themselves, regardless of what is located there. So, for instance, we might say that a spacetime point x actually causes spacetime point y to exist. Call this type of actual causation: elemental causation, since it holds between the basal elements of a theory, in this case between the spacetime points.
Elemental causation is best demonstrated using the interventionist framework outlined below. The basic idea, though, is this: elemental causation occurs when, were one to ‘remove’ an element x this would make a difference to whether another element y exists, or to the probability of y existing (p.4).
This focuses attention on the elements of CST and the theory’s growth dynamics: this is properly characterized as that of a process where existing elements cause new elements to come into being. I note that sometimes the authors’ language invokes the notion of productive (rather than dependence/difference-making) causation:
At the fundamental level, there would need to be actual causal relations linking causal set elements. Where, and this is important, those actual causal relations hold between the elements themselves (rather than whatever is located at those elements), so that elements literally bring other elements into existence (p.4, my emphasis).
One thing that occurs to me is that a the more familiar notion of “material causation” occurring between entities in spacetime would also be a derivative notion: It could be that patterns in the elements at the CST level give rise to material causation “in” spacetime.
They foreshadow their conclusion:
We propose a limited answer to the question of whether there is elemental causation in CST. We argue for the affirmative, that there is elemental causation in CST, however our argument is conditional, in this sense: there is elemental causation in CST on the condition that a particular interpretation of the dynamics is adopted. Specifically, an interpretation in which the growth process in the standard dynamics corresponds to a real physical process (pp. 5-6).
Again, they mention the skepticism some have felt with CST growth dynamics (associating it with “passage”): I’ll have to look at the references to get an appreciation for the objection.
Section 3. Causal Set Theory
The basic kinematic structure of CST (partially ordered set of elements) is described and then the dynamics of adding elements (one at a time). This process is stochastic, and a new element may be linked to one or more parent elements or may be unlinked.
Another thing of note: the dynamics as described have a feature that may be important to its role as a possible path to a theory of quantum gravity: that is what the authors (relying on previous CST work) call discrete general covariance (pp. 10-12). Multiple paths might be taken to arrive at a particular causal set, and the details of the arrangement of elements along specific paths are not relevant:
As noted, this is interpreted to mean that there is no physical significance to how we might order elements in the final causal set in terms of when they were added via the dynamics, since it is equally likely that the final causal set was birthed from multiple paths that, under a choice of gauge, involve different orderings of elements (p.10).
The reason I highlight this is that it calls into question whether particular elements in a set are distinguishable from each other. In the kinematics, the elements of a causal set have nothing to distinguish them other than their relations to other elements. When we add the dynamics, it might appear that elements recover something more unique in virtue of their role in building a particular historical growth trajectory, but the covarience stipulation calls this into question (more on this below).
Section 4. Interventionism
The authors describe the interventionist framework. Then they raise two worries about applying it here. One is that CST is a model of an entire universe, so the intervention qua additional cause would come from outside the universe. Secondly, interventionism is not a theory of causation from scratch: intervention is a causal notion, so to add an intervention to assess its impact assumes the system is already causal in the same sense. In the context of this paper (which wants to assess the presence of “actual causation” in CST), this looks question-begging (p. 14). To address this, the authors want to use the notion of a “setting intervention” (Woodward’s term), which is one that doesn’t invoke an intervention as a “possible cause” coming from outside the system. (See discussion on pp. 14-16, including references to possible criticisms of this move in the context of physics).
Comment: I’m not sure there is really anything important at stake with regard to the discussion of “setting interventions” vs. “possibility-constrained interventions”. (There is more discussion of this in the section on objections—see the fifth objection beginning on p. 29). From my perspective, the key issue is that both introduce counterfactual scenarios, as opposed to basing the analysis on what is actually happening in the universe. This gives you a way to discover/reveal a particular sort of causal relation, but, and this is my main reason for preferring productive causation, it is not a viable metaphysical theory of causation for growth dynamics. That would have to be one that builds a causal foundation into the fabric of the actual universe.
Section 5. Application
The section begins with more discussion re: the controversy in the literature over whether sequential growth should be considered a real physical process (p. 17). Authors will first assume that this is the case. They apply an intervention to the second stage of a three-stage toy model (a “switch”) and show that it makes a change to probabilities for the third stage, indicating causation. And it is “elemental causation” because one of the differences relates to creation of a “new” unlinked element in the variation.
Next, they reconsider whether growth is a real physical process, focusing on the implications of the discrete general covariance assumption, where the individual token instances of a type of situated element doesn’t matter, and hence can be interpreted as lacking physical significance. This may imply that the differences seen in individual stages shouldn’t be taken seriously, because some will wash out once we have a complete (infinite) causal set:”
The problem generalises in an uncomfortable way: there appears to be no fact of the matter as to what exists at the nth stage for any finite stopping point, because the probability measure of causal sets is only well-defined in the limit. It is thus only once the dynamics is run to infinity that the causal set ‘snaps’ into place (p.22).
But general covariance doesn’t seem to wash out all the features of growth stages in the set. First, the authors discuss Wüthrich and Callendar’s (2017) work and the notion of “posts” – see pp. 22-23. Some causal sets feature an element that is connected to all others via ancestry—this is called a post. When a post is reached, the set’s form can be assessed without waiting for infinity. More importantly perhaps for the present analysis, the authors note that cardinality (number of elements in a set) is well defined at each (“generational”) step even if the features of the set are not all determinate. Focusing on this feature, the authors discuss an intervention that prevents the addition of a new element at a stage (23-24), cutting off growth. This intervention obviously makes a difference and hence establishes the presence of actual elemental causation.
The authors consider a number of objections to all of this (pp. 25-33). Of particular interest is the fifth objection (beginning on p. 29) which discusses how to think about the interventionist counterfactuals, and whether this is an appropriate application. The authors stress that it is appropriate in the context of investigating the CST framework itself:
We are drawing conclusions only about causal set theory itself. Matters would be different if we were trying to draw conclusions about the world, or about another theory beyond CST. But that’s not what we’re doing (p. 31).
The paper ends with a brief recapitulation in a concluding section (p.33).
Comments:
This is a helpful paper that draws out in greater depth the sort of causal goings-on in CST (including distinguishing it from the causal connectibility structure of GR). My interests are more about seeing whether a causal process metaphysics can be adapted for CST, and thus goes beyond this work, but is a great spur to thinking about the topic.
In the prior post about CST I expressed my sentiment that a production theory is a more natural fit for CST. I mentioned that one way to put a production gloss on the CST growth model would be to endow elements in each generation with the causal powers/propensities to create a new element. But a better way to look at it (providing a closer fit to extant causal process theories) is to stipulate that the elements in the set correspond to the interaction events between processes. The links between elements would correspond to the connecting causal processes (which propagate their powers between interaction events). In this way of thinking, the causal set is a formal physical model of the underlying causal ontology—one that abstracts from some of the metaphysical details. On the other hand, I note that there are two possible inconsistencies standing in the way of this interpretation. The first involves the creation of new elements, and the second has to do with the lack of distinguishability of the elements.
The creation of brand-new elements in each step of CST growth appears to require something not explicitly included in my previous presentations of causal process theory: that of something like fissioning events (one additional process emerges from an interaction). This seems OK. However, the notion of creating new elements without connections to prior elements is very hard to reconcile with a network of causal processes as I’ve described it. But here it is interesting that other work on CST has explored dynamics where such “orphan” elements are excluded. This is connected to the idea that causal sets will typically contain “posts”, and if we only focus on the dynamics of elements connected to posts, we get interesting models that may connect to cosmological scenarios. In the paper cited below, the authors call the restricted version of dynamics (which requires parentage) “originary dynamics”:
Note that the effective dynamics following a post comes with a caveat: each element is required to be related to the post, by definition. Therefore we find an originary dynamics, for which the possibility of being born unrelated to any other element is excluded, and all remaining probabilities are normalized correspondingly. Thus the probabilities of an originary dynamics are equal to those of an ordinary CSG model, conditioned on the event that the newborn element connects to at least one other element (Ahmed and Rideout, 2010, preprint version, p.4).
So, perhaps one could just exclude the notion of new elements with no relations as superfluous or arguably non-physical.
The second seeming inconsistency is the lack of significance associated with the becoming of particular individual elements given the assumption of discrete general covariance (although the growth process complicates this assertion). In the kinematics, I don’t worry about the fact that elements have nothing to distinguish them apart from their relations within the set: to the extent this seems to differ from the what is implied by the underlying theory, that can be attributed to the fact that the causal set is an abstract model of the ontology. However, in the dynamics, the stipulation that the path taken to a given causal set doesn’t matter seems to conflict from causal process theory as I have presented it. There, while the processes emanating from a set of interaction events (modeled as causal set n) may have a joint probabilistic disposition toward various outcomes in the next stage, only one of these outcomes (which becomes the set of new interaction events or causal set n+1) is realized at each step. So, in the case of evolution spanning multiple generations, a determinate path between start and finish is implied. Perhaps one way to think about general covariance is to by analogy with the indeterminacy of quantum systems prior to measurements. At least under some interpretations of quantum mechanics, we consider all of the possible paths of a particle (prior to measurement) to exist in some sense, but in superposition. This requires further thought, and it is a reminder that we unfortunately lack a genuinely quantum version of CST that has wide endorsement from researchers in the field.
Finally, I should mention that a presumed benefit of associating causal process ontology with CST is the idea of uniting this with the ontology proposed for matter/energy. As I alluded to above, perhaps this opens the door to see the latter as a larger scale pattern in the much more finely-grained CST network.
References
Ahmed, M., Rideout, D. 2009. Indications of deSitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets. Arxiv preprint. https://arxiv.org/abs/0909.4771
Baron, S., Le Bihan, B. 2025. Causal Set Theory is (Strongly) Causal. Foundations of Physics 55, 63. https://doi.org/10.1007/s10701-025-00875-w