Dear Robert,
I thought you might be interested in the following idea I posted on George Ellis's thread. Since you also are interested in "nonmanifold models that emphasize the role of causality," I thought I'd copy the idea here.
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After initially struggling with the idea, I've been thinking a bit about how your [George's] top-down causation idea might look from the perspective of nonmanifold models of fundamental spacetime structure that emphasize the role of causality. It seems that top-down causation might provide an interesting new perspective on such models. For definiteness and simplicity, I use Rafael Sorkin's causal sets approach as an example.
Causal sets, as currently conceived, are by definition purely bottom-up at the classical level. Causality is modeled as an irreflexive, acyclic, interval-finite binary relation on a set, whose elements are explicitly identified as "events." Since causal structure alone is not sufficient to recover a metric, each element is assigned one fundamental volume unit. Sorkin abbreviates this with the phrase, "order plus number equals geometry." This is a special case of what I call the causal metric hypothesis.
In the context of classical spacetime, top-down causation might be summarized by the statement, "causal relationships among subsets of spacetime are not completely reducible to causal relations among their constituent events." In this context, the abstract causal structure exists at the level of the power set of classical spacetime, i.e., the set whose elements are subsets of spacetime. Discrete models very similar to causal sets could be employed, with the exception that the elements would correspond not to events, but to families of events. Two-way relationships would also come into play.
Initially this idea bothered me because of locality issues, but such a model need not violate conventional classical locality, provided that appropriate constraints involving high-level and low-level relations are satisfied.
This idea is interesting to me for the following reasons.
1. The arguments for top-down causation presented by you [George] and others are rather convincing, and one would like to incorporate such considerations into approaches to "fundamental theories," particularly those emphasizing causality.
2. One of the principal difficulties for "pure causal theories" is their parsimony; it is not clear that they contain enough structure to recover established physics. Top-down causation employed as I described (i.e. power-set relations) provides "extra structure" without "extra hypotheses" in the sense that one is still working with the same (or similar) abstract mathematical objects. It is the interpretation of the "elements" and "relations" that becomes more general. In particular, the causal metric hypothesis still applies, although not in the form "order plus number equals geometry."
3. There is considerable precedent, at least in mathematics, for this type of generalization. For example, Grothendieck's approach to algebraic geometry involves "higher-dimensional points" corresponding to subvarieties of algebraic varieties, and the explicit consideration of these points gives the scheme structure, which has considerable advantages. In particular, the scheme structure is consistent with the variety structure but brings to light "hidden information." This may be viewed as an analogy to the manner in which higher-level causal structure is consistent with lower-level structure (e.g. does not violate locality), but includes important information that might be essential in recovering established physics.
4. As far as I know, this approach has not yet been explicitly developed.
I'd appreciate any thoughts you might have on this.
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I'd appreciate your thoughts on this too! Take care,
Ben
