# Does causality always require an arrow of time?

For empirical facts, it seems obvious that causality requires a time flow for the concept to make sense: A causes B implies that A happened before B.

Is it ever possible to have a causal relationship without a flow of time? Presumably this is impossible for empirical facts, as an effect preceding a cause goes against the laws of physics. Is this always true?

But what about relations of ideas? Is it possible to have a causal relationship between two ideas?

Colloquially we often speak of ideas having causal relationships:

• Equation B is a consequence of theorem A.
• Proposition Y is true because of proportion X.

Are these true causal relationships? Or is it merely us projecting our temporal thought processes onto the ideas and propositions we are analyzing and hence seeing causality where there is none? Can relations of ideas have causal relationships?

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The term "cause" necessitates that one thing precedes another. But it sounds like you may be asking whether cause-and-effect are just illusions or coincidences. – Ben Piper Jan 4 at 22:21

Nancy Cartwright's definition, applied to mind-independent reality, does not reference time:

C causes E if and only if C increases the probability of E in every situation which is otherwise causally homogeneous with respect to E. (Causal Laws and Effective Strategies, 423)

The technical definition of "causally homogeneous with respect to E" is articulated on p423ff. I myself encountered the above definition in Cartwright's How the Laws of Physics Lie, which contains a slightly expanded version of the essay I've cited.

Switching to mind-dependent reality, you might want to consider using the term 'entailment', which can be seen as more general than 'causation'. Mathematical biologist Robert Rosen looks at the issue extensively in Life Itself, and uses the mathematics of category theory to rigorously describe the relationship between entailment in a model of reality and causation in reality. There is also some interesting discussion of whether causation/entailment exists in the model or also in reality in the beginning of Martin Hollis' Models of Man.

One way to consider the difference between logical entailment in formal systems and causation in reality is to recognize that not all conceptions of causation in reality are necessitarian. In contrast, logical entailment in a formal system is necessitarian. Given that there does not seem to be a link between necessitarianism and time, I shall say no more on this topic.

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The answer depends on the definition of "causal", but traditional metaphysics did not require causal relation to be temporal. The most famous counterexample is the relation between God and the world in Christian theology modeled on the relation between One and the world in the neo-Platonism of Plotinus. God or One are eternal/atemporal and do not even exist, their being logically precedes existence and time that flow from it, i.e. they cause the temporal world to exist. Kant modeled the relation between his atemporal self, which exercises free will, and empirical self inserted into phenomenal causal chains that play out in time on this theological causality, but he would not use the word "cause" in this context.

There is an interesting application of this atemporal causality in modern physics, the Hartle-Hawking no-boundary cosmology misrepresented in popular literature as having atemporal piece "before" the Big Bang. Butterfield and Isham in On the Emergence of Time in Quantum Gravity explain that this is not the case:"One often sees a picture in which a cone-like spacetime structure (representing a cosmological solution of classical general elativity) is attached to a spherical shape that represents a Euclidean 4-manifold. This erroneously suggests that the bottom sphere is straightforwardly earlier than the classical cosmology represented by the open cone in the top half of the figure. But the 4-manifold is not earlier: there is no temporal relation between the two halves represented in the figure (or their parts)!" The relation is more akin to the theological causality with God replaced by a quantum ensemble of Euclidean 4-manifolds, and the world by Lorentzian spacetime of Big Bang cosmology implemented by their superposition. This is often misleadingly expressed as "tunneling from nothing", with "nothing" referring to the Euclidean "pure space", and "tunneling" to "moving from a time variable that is a real number to one that is purely imaginary". An incomplete classical analogy would be atoms in a solid "causing" elastic waves when they follow a periodic pattern of motions. The relation between motions and waves is not temporal, they are co-extensive, but in the no-boundary cosmology the time itself emerges from an atemporal substrate, just like waves from atoms.

While temporality can perhaps be avoided causality does seem to require some sort of partial order relation to be meaningful. In the cases of creation and emergence time is replaced by a "flow" between different "levels" of ontology. This being said, even if one admits atemporal causality it does not have to apply to every case of logical entailment. Indeed, on the Platonist reading there are no causal relations between ideal forms, they are at the same level, independent and co-eternal, hence mathematical derivations do not represent any kind of causality. Causality of derivations would not conform with the usual mathematical intuition either, if we have two equivalent mathematical facts (e.g. being even and being a sum of two primes) what sense would it make to say that one is caused by the other?

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"replaced by a "flow" between different "levels" of ontology" fair enough. But then how does one differentiate between "artificial flow" imposed by our thought process, vs "inherent flow" which indicate a true causality? – Alexander S King Jan 5 at 0:24
@Alexander S King The "flow" between different levels may well be "inherent" as well, certainly theologians view creation from nothing as independent from our thought processes, I think the same goes for Hartle-Hawking. According to Kant, even temporal "true causality" is imposed by our mental faculties, and he agrees with Hume that there is none other to be had. I think the distinction between inherent and imposed depends on metaphysics one adopts, and so will the methods of differentiation. To empiricists the question is moot except within a model describing "reality" and "observer". – Conifold Jan 5 at 1:16

The issue is controversial. In particular, imagine we live in a universe with cyclic time, or imagine that the geometry of space time is such that some processes go backward in time. Then causes could precede their effect. There are also interpretations of quantum mechanics that involve retrocausal effects.

However we have deep intuitions that causes precede their effects. A simple argument to express that intuition is that if effects preceded their causes then the cause could be undone and their wouldn't be a causal relationship so we'd have a potential contradiction.

Regarding mathematical statements it is indeed a projection of our thought process on deductibility relations: there is no causal relations between premises and their consequences, but the fact that we use causal language is an interested feature that were analyzed by some authors (sorry I don't remember the names).

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When you ask if "causality" requires an "arrow of time," I believe it is not impertinent to say, the opposite is more nearly the case.

The Newtonian, "billiard ball" picture of causation is reversible. Run the events backwards and you could not tell the difference. This is why such mechanical causality can be captured in mathematics, where there is no friction, disorganization, or entropy.

Entropy or the "Arrow of Time" is generally taken as the peculiar directionality of time as increasing "disorder" from some "point of view" and the apparent fact that such a point of view can never actually be eliminated from the picture. What occurs cannot be reversed. The "arrow" does not, cannot possibly, return to the bow. If this is "really" the case, and we assume it really is, then no "cause" is reversible or perfectly repeats itself, and "causation" appears pragmatic or nominalist.

You also ask if this is "just an idea." Teleology and the four Aristotelean "causes" seem to be making a comeback in "emergence," etc., against the old Newtonian, billard-ball model. If, say, "strange attractors" or "black holes" or even "desires" have some causal force, do they act before or after the "effect?" (Confusion of A series, B series time?)

Hegel seems to take a somewhat nominalist or at least ambiguous view of mechanical "causation." He rightly pointed out that causation is retrospectively selective. One cannot identify a "cause" until one has identified the "effect." So in conscious representation, effects always precede causes.