Robert Thibadeau, Ph.D.
Another Saturday Afternoon on the Internet
Conventional cosmology states the argument that a universe that is always becoming more disordered, from a more ordered state, is a good way to talk about the arrow of time, because successive states move from more ordered states to less ordered states by probability alone. The less ordered state is simply vastly more probable than the ordered state.
Physics is uncomfortable with causality as a law because equations, or symmetries, are not assertions of causal relationships, but of equality relationships. This reasoning bias exists in favor of mathematics despite the fact that theory verification is based on establishing causality. Relations that assert "X causes Y" are fundamental inequalities, or asymmetries.
But the human mind organizes nearly all experience and knowledge in causal terms. All but a handful of verbs in any language contain a causal entailment. Non-causal verbs in English include "is" and "has." But even "contains" can be used causally, as in "The guards contained the prisoners" ('the guards caused the prisoners to be contained'). This giant disconnect between how humans organize all their perceptions and language, and how physics describes reality is interesting to say the least.
The interesting question is whether causality exists in nature, in physics, independently of our perception and thought. This is a relationship between how people perceive and organize their experiences and how the world (may) actually work. The suggestion to be made here is in accord with theorists who have argued that the human mind is organized only to reflect invariants in reality. The conclusion is that causality is in fact a local property of inexorable movement from lower to higher entropy. Causality, however, is a mental recognition that this movement locally reverses, in physical reality, from higher to lower. That brief reversal, for a specific time-space nexus, causes a human to focus on a causal event, an event that makes a difference in surviving. An event worth remembering.
Contrafactuals and Implication
So, how do we make this connection? We begin with a law of thought, language, and perception noted by Lewis (1973) and Simon and Rescher (1966), namely that causality can be found in any conditional whose contrafactual cannot be contraposed. The example commonly used is "if it rains, the wheat grows." This conditional is perceived as causal. That perception is precisely, and only, because its contrafactuals, "it did not rain" and "the wheat did not grow" cannot be contraposed. It is not true that "if the wheat did not grow, then it didn't rain."
The contraposition of the contrafactual is not true. Other conditionals, such as "if he moves into the room, then he is in the room" can be successfully contrafactually contraposed, "if he is not in the room, then he did not move into the room." This latter case is a pure symmetry. The former is a pure causation. Both are conditionals. If the wheat did not grow, it is indefinite if it rained. If he is not in the room, it is indefinite whether he moved into the room. But only one contraposed counterfactual is false: it is not true that if the wheat did not grow, then it didn't rain, versus it is true that if he is not in the room then he didn't move into the room.
Simon and Rescher (1966) rejected this type of analysis and proposed an alternative, and they argued more fundamental, concept that cleverly showed how a system of equations could capture the asymmetry and the universal sense of causality. Rather, in certain systems, solving some equations must precede the solution of others. This they called causality. To quote one of their examples:
"
" (Simon & Rescher 1966, pp 326-7)
I now argue that this is directly a serialization in uncertainty reduction since the number of unsolved variables in the system is reduced and each unknown variable naturally has a number of possible values until a value is derived. If we assume that, in general, all variables have an equal number of alternative states and all states have an equal probability, we can express causality directly as a reduction in entropy in the system described by the particular system of equations allowed by Simon and Rescher (1966). They call such a system a "Complete Structure" and provide two definitions for it:
"Definition 1: A structure is a set of m functions involving n variables (n >= m), such that:
(a) In any subset of k functions of the structure, at least k different variables appear.
(b) In any subset of k functions in which r (r >= k) variables appear, if the values of any (r - k) variables are chosen arbitrarily, then the values of the remaining k variables are determined uniquely. (Finding these unique values is a matter of solving the equations for them).
Definition 2: A structure is self-contained if it has exactly as many functions as variables." (Simon and Rescher, 1966, pp. 324-5).
A Complete Structure must obey definition 1 and 2, but it is in 1(b) that the serialization of outcomes occurs, and here is where causation emerges. It is also, true, I believe, that this serialization to a completely uniquely determined result is always a stepwise reduction in entropy.
Thus, we find E = mc2 does not state causation, although the system of equations that derive and m, and perhaps c, may lead to a series of causal steps in computing the energy of a particular nuclear explosion. Without the background system of equations that must be solved serially, there is no causation that can be enumerated from E = mc2. There is no boom.
The difference between causality and implication is that one, causation, locally reduces entropy, while the other, logical implication or equality, does not. In effect, the condition, the cause, reduces the entropy of the consequent, the result. Implication does not affect or reflect entropy.
Simon and Rescher rejected counterfactuals as irrelevant, but actually wind up finding them relevant, I believe. Simon & Rescher (1966) show that when you have an analysis of dichotomous (true-false) variables that solves serially you capture causality again. It is a well recognized rule for material conditionals, "if X then Y", that if X is true, then Y is true. Also, if X is false, then Y may be true or false. Causal conditionals are those where if Y is false, X must be indeterminately true or false. If Y is false then X may be true ('if the wheat did not grow, it may have rained'). Such indeterminism ('in whether the wheat grew or did not grow in the presence of rain') is precisely a form of entropy: If X caused Y, there is a reduction in entropy ('the wheat grew' as opposed to being indeterminately growing or not growing).
In assessing entropy gain or loss in any situation, the Boltzmann model says to count the number of microstates consistent with the macrostate. If the number of microstates increases, entropy increases. So think of causality as a brief and localized change from a progressive disordering to an ordering within Simon and Rescher Complete Structures. Perhaps huge numbers of them as atoms causally hit atoms. Causality is when the normal progression of entropy is locally reversed within Complete Structures. Such reversals would and should be occurring all the time. So, to see them occurring is not a surprise. It is also not a surprise to see them at microscales and macroscales (and perhaps even quantum scales?). Macroevents that order, particularly predictive events that order, should be of high survival value. They allow an organism to learn in a field of disorder where learning is otherwise not possible.
So, how do we measure entropy change? The easy way is to say that a prior event, an event at t-1, somehow reduces the uncertainty of an event at t. Why is this not the same as a conditional that can be contrafactually contraposed? It is precisely because, in that case, the condition fully determines the result. The condition, itself "a surprise", is the instance of uncertainty reduction. But in that case, nothing 'caused' or 'predicted' it. However, once the condition was observed the consequent was inevitable. In causality, the consequent is not inevitable. When the consequent is observed it is confirmed as caused by the conditional. The entropy reduction is complete, but only on observation of the consequent. Indeed, the perceptual 'law' of Simon and Rescher, says that the consequent is recognized as ordered precisely because the conditional occurred, but not when the conditional occurred. The human (or animal?) question is always because the conditional does not determine the consequent until the consequent occurs. Otherwise, the contrafactual, the falsification of the consequent, can be contraposed, if the consequent occurs, then the conditional occurred.
This sounds just like experimentation. When we observe the result, we make the conclusion that X caused Y. We do not make the conclusion simply in the presence of the condition although we may compute an expectation of the conclusion if our theory is good.
This is the essence of causality. Causality is perhaps a law of nature. Our minds did not invent it. It is not a nuisance to physics. It is the essence of an observed change to lower entropy within a necessarily serialized set of (what amount to) entropy computations. Predictability itself lowers entropy, as has been shown in information theoretic treatments of entropy. However the lower entropy state is brief and can be rapidly expanded to non-entropic movement.
So what do we say? We characterize the entropy measurement region as any selected time-space region where confirmation or realization of the consequent is dependent on a necessary serialization of antecedents defined within a Complete Structure. It is inherently asymmetric. It inherently moves only forward although, it need not move forward in time. Causality is a successive reduction in localized uncertainty where the localization is completely captured by a particular Simon and Rescher Complete Structure.
If we say this, where does the increase in entropy come from? It comes from where it came before. Despite the practically uncountable number of causal events happening all about us, despite all these local reductions in uncertainty, causality itself alters states probabilistically precisely because the antecedent conditions mix prior states. The entropy of the larger system can only increase. The number of equations that need to be solved increases as the number of cause-effect antecedent events increases. An, in fact, if this is right, without causation the entropy can't change, in either direction.
Is it possible to observe a local decrease in uncertainty without being able to observe causality? The answer would be no.
Notes
1. Since this analysis is intended to relate to physical reality and human perception and thought, the analysis is strictly applied to denotational semantics. In effect, we are looking for where "causation" is denoted. Clearly it is possible to develop semantic or logical systems that do not apply denotationally to physical reality where many of the observations made here are utterly invalid.
2. In referring to related work, "Contrafactual" and "Counterfactual" are used interchangeably.
References
Simon, Herbert and Nicholas Rescher. 1966. "Cause and Counterfactual." Philosophy of Science 33: 323–40. (Available through JSTOR)
Lewis, David. (1973) "Causality." The Journal of Philosophy 70:556-567.
*----. (1979) "Counterfactual Dependence and Time's Arrow" Noûs 13: 445-476.
*----. (2000) "Causation as Influence" The Journal of Philosophy 97: 182-197.
http://en.wikipedia.org/wiki/Causality -- Look at my entry for Derivation Theories for a more precise presentation of Simon and Rescher 1966.
http://en.wikipedia.org/wiki/Entropy
Also, it was an audible.com version of Fabric of the Cosmos by Brian Greene that led me to the association between entropy and Simon and Rescher's work. I had known Simon and Rescher's work since the early 1970s and always felt it was much more important than people had given credit.