Probabilistic propensities and the Aristotelian view of time

Consider an item x with a half-life of one hour. Then over the period of an hour, it has a 50% chance of decaying, over the period of a second it only has a 0.02% chance of decaying. Imagine that x has no way of changing except by decaying, and that x is causally isolated from all outside influences. Don’t worry about Schroedinger's cat stuff: just take what I said at face value.

We are almost sure that x will one day decay (the probability of decaying approaches one as the length of time increases).

Now imagine that everything other than x is annihilated. Since x was already isolated from all outside influences, this should not in any way affect x’s decay. Hence, we should still be almost sure that x will one day decay. Moreover, since what is outside of x did not affect x’s behavior, the propensities for decay should be unchanged by that annihilation: x has a 50% chance of decay in an hour and a 0.02% chance of decay in a second.

But this seems to mean that time is not the measure of change as Aristotle thought. For if time were the measure of change, then there would be no way to make sense of the question: “How long did it take for x to decay?”

Here is another way to make the point. On an Aristotelian theory of time, the length of time is defined by change. Now imagine that temporal reality consists of x and a bunch of analog clocks all causally isolated from x. The chances of decay of x make reference to lengths of time. Lengths of time are defined by change, and hence by the movements of the hands of the clocks. But if x is causally isolated from the clocks, its decay times should have nothing to do with the movements of the clocks. If God, say, accelerated or slows down some of the clocks, that shouldn’t affect x’s behavior in any way, since x is isolated. But an Aristotelian theory of time, it seems, such an isolation is impossible.

I think an Aristotelian can make one of two moves here.

First, perhaps the kinds of propensities that are involved in having an indeterministic half-life cannot be had by an isolated object: such objects must be causally connected to other things. No atom can be a causal island. So, even though physics doesn’t say so, the decay of an atom has a causal connection with the behavior of things outside the atom.

Second, perhaps any item that can have a half-life or another probabilistic propensity in isolation from other things has to have an internal clock—it has to have some kind of internal change—and the Aristotelian dictum that time is the measure of change should be understood in relation to internal time, not global time.

God, probabilities and causal propensities

Suppose a poor and good person is forced to flip a fair and indeterministic coin in circumstances where heads means utter ruin and tails means financial redemption. If either Molinism or Thomism is true, we would expect that, even without taking into account miracles:

1. P(H)<P(T).

After all, God is good, and so he is more likely to try to get the good outcome for the person. (Of course, there are other considerations involved, so the boost in probability in favor of tails may be small.)

The Molinist can give this story. God knows how the coin would come out in various circumstances. He is more likely to ensure the occurrence of circumstances in which the subjunctive conditionals say that tails would comes up. The Thomist, on the other hand, will say that God’s primary causation determines what effect the secondary creaturely causation has, while at the same time ensuring that the secondary causation is genuinely doing its causal job.

But given (1), how can we say that the coin is fair? Here is a possibility. The probabilities in (1) take God’s dispositions into account. But we can also look simply at the causal propensities of the coin. The causal propensities of the coin are equibalanced between heads and tails. In addition to the probabilities in (1), which take everything including God into account, we can talk of coin-grounded causal chances, which are basically determined by the ratios of strength in the causal propensities. And the coin-grounded causal chances are 1/2 for heads and 1/2 for tails. But given Molinism or Thomism, these chances are not wholly determinative of the probabilities and the frequencies in repeat experiments, since the latter need to take into account the skewing due to God’s preference for the good.

So we get two sets of probabilities: The all-things-considered probabilities P that take God into account and that yield (1) and the creatures-only-considered probabilities P_c on which:

2. P_c(H)=P_c(T)=1/2.

Here, however, is something that I think is a little troubling about both the Molinist and Thomist lines. The creatures-only-considered probabilities are obviously close to the observed frequencies. Why? I think the Molinist and Thomist have to say this: They are close because God chooses to act in such ways that the actual frequencies are approximately proportional to the strengths of causal propensities that P_c is based on. But then the frequencies of coin toss outcomes are not directly due to the causal propensities of the coin, but only because God chooses to make the frequencies match. This doesn’t seem right and is a reason why I want to adopt neither Molinism nor Thomism but a version of mere foreknowledge.

More on Molinism and stochastic processes

In earlier posts and comments, here and on prosblogion, Mike Almeida and I have been discussing problems with Molinism and stochastic processes.

Here's a way to put a variant of the problem (this may well duplicate some of Mike's ideas). Let C be the following set of circumstances: a fair indeterministic coin is tossed, with a machine set up so that if the coin landed heads, then laws of nature specify that the machine would cause all creatures in existence suffer excruciating and undeserved pain for eternity.

We can now do two different calculations. Let G be the claim that omnipotent, omniscient and perfectly good God necessarily exists. On the one hand, P(Heads|C)=1/2 (because the coin is fair). On the other hand, P(Heads|C and G) is less than 1/2. For such a God would be unlikely to allow C to be actualized unless he knew that the counterfactual C→tails is true. He might of course be planning to miraculously intervene after the machine activates, and so P(Heads|C and G) is non-zero but it seems to be part of divine providential goodness to avoid having to intervene miraculously, but surely P(Heads|C and G) is less than 1/2.

But now we actually have a contradiction. For the probabilities in question seem to be objective probabilities, and when we're talking of objective probabilities, P(G)=1, since G is a necessary truth. Hence, 1/2 > P(Heads|C and G)=P(Heads|C)=1/2. In other words, 1/2 > 1/2, which is absurd.

Therefore, we must reject one of the two probability claims. In particular, it seems, we need to reject P(Heads|C)=1/2. But this means that given theism, we cannot consider the probabilities that come from empirical study to be the genuine objective probabilities governing the events. Granted, in the case above, we were talking of a catastrophic case. But presumably even if the consequences of heads are somewhat bad, P(Heads|C and G) will still be somewhat less than 1/2.