God and chancy infinite causal regresses

Suppose that a dod is a critter that chancily, with probability 1/2, causes one offspring during its life. The lifespan of a dod is one year. Further, imagine that like Sith, there are only ever one or two dods at a time, because each dod dies not long after reproducing, and if there were two or more mature dods at once, they’d fight to the death.

Now, imagine we have an infinite regress of dods, because each dod comes from an earlier dod. This would be hard to believe! After all, at any time at which we have a dod, we should be extremely (infinitely?) surprised that the dods haven’t died out yet. After all the probability that, given a dod at some time that there would be a dod in n years exponentially decreases with n.

Assuming causal finitism is false, it seems God could intentionally create an infinite regress of dods. But what would that look like? Here’s one story. God overrides the chances and directly and intentionally creates a backwards-infinite (and maybe even forwards-infinite, if he so chooses) sequence of dods. In that case, within that sequence the 1/2 chance of dod reproduction plays no explanatory role whatsoever. It seems we have occasionalism or a miracle or both. In any case, it does not appear that we actually have an infinite causal regress of dods in this case—the causation between dods, with its 1/2 chance, seems not to have any explanatory role. So the “overriding” story doesn’t work.

The other option is the Thomistic story. God doesn’t override chances. Instead, through primary causation, God concurs in creaturely causation and makes the finite cause produce its effect in such a way that the finite cause is fully acting as an indeterministic cause (this goes along with a view on which God can make us freely and indeterministically choose things). But this is very strange. For what explanatory role does the 1/2 in the chancy causation play? Assuming God wanted there to be an infinite sequence of dods, he could do exactly the same thing if the chance were 1/10 or 9/10 or even 1. It seems that the dod reproduces if and only if God intends the dod to reproduce, and whether God intends the dod to reproduce seems to have nothing to do with the “1/2” in the dod’s reproductive probabilities—it’s not plausible that God has probability 1/2 of intending each given dod to reproduce. And if God had probability 1/2 of intending each given dod to reproduce, how could he intentionally ensure that there ever are any dods, since the probability that God has infinitely many of these individual-dod-reproduction intentions is zero.

So we have problems. This gives further evidence that theism implies causal finitism.

Simultaneous causation and occasionalism

In an earlier post, I said that an account that insists that all fundamental causation is simultaneous but secures the diachronic aspects of causal series by means of divine conservation is “a close cousin to occasionalism”. For a diachronic causal series on this theory has two kinds of links: creaturely causal links that function instantaneously and divine conservation links that preserve objects “in between” the instants at which creaturely causation acts. This sounds like occasionalism, in that the temporal extension of the series is entirely due to God working alone, without any contribution from creatures.

I now think there is an interesting way to blunt the force of this objection by giving another role to creatures using a probabilistic trick that I used in my previous post. This trick allows created reality to control how long diachronic causal series take, even though all creaturely causation is simultaneous. And if created reality were to control how long diachronic causal series take, a significant aspect of the diachronicity of diachronic causal series would involve creatures, and hence the whole thing would look rather less occasionalist.

Let me explain the trick again. Suppose time is discrete, being divided into lots of equally-spaced moments. Now imagine an event A₁ that has a probability 1/2 of producing an event A₂ during any instant that A₁ exists in, as long as A₁ hasn’t already produced A₂. Suppose A₁ is conserved for as long as it takes to produce A₂. Then the probability that it will take n units of time for A₂ to be produced is (1/2)^n + 1. Consequently, the expected wait time for A₂ to happen is:

• (1/2)⋅0 + (1/4)⋅1 + (1/8)⋅2 + (1/16)⋅3 + ... = 1.

We can then similarly set things up so that A₂ causes A₃ on average in one unit of time, and A₃ on causes A₄ on average in one unit of time, and so on. If n is large enough, then by the Central Limit Theorem, it is likely that the lag time between A₁ and A_n will be approximately n units of time (plus or minus an error on the order of n^1/2 units), and if the units of time are short enough, we can get arbitrarily good precision in the lag time with arbitrarily high precision.

If the probability of each event triggering the next at an instant is made smaller than 1/2, then the expected lag time from A₁ to A_n will be less than n, and if the probaility is bigger than 1/2, the expected lag time will be bigger than n. Thus the creaturely trigger probability parameter, which we can think of as measuring the “strength” of the causal power, controls how long it takes to get to A_n through the “magic” of probabilistic causation and the Central Limit Theorem. Thus, the diachronic time scale is controlled precisely by creaturely causation—even though divine conservation is responsible for A_i persisting until it can cause A_i + 1. This is a more significant creaturely input than I thought before, and hence it is one that makes for rather less in the way of occasionalism.

This looks like a pretty cool theory to me. I don’t believe it to be true, because I don’t buy the idea of all causation being simultaneous, but I think it gives a really nice.

Explanation and understanding

In the 1960s, it dawned on philosophers of science that:

1. Other things being equal, low-probability explanation confers equally good understanding as high-probability explanation.

If I have a quantum coin that has a probability 0.4 of heads and 0.6 of tails, and it yields heads, I understand why it yielded heads no less well than I would have had it yielded tails—the number is simply different.

On the other hand, the following thesis (which for years I’ve conceded to opponents to low-probability explanations):

2. Other things being equal, low-probability explanations are less good than high-probability ones.

Finally, add this plausible comparative thesis:

3. What makes an explanation good is how much understanding it confers (or at least would confer were it true)

which plausibly fits with the maxim that I’ve often been happy to concede that the job of an explanation is to provide understanding.

But (1)–(3) cannot all be true. Something must go. If (2) goes, then Inference to Best Explanation goes as well (I learned this from Yunus Prasetya’s very recent work on IBE and scientific explanation). I don’t want that (unlike Prasetya). And (1) seems right to me, and it also seems important to defending the Principle of Sufficient Reason in stochastic contexts.

Reluctantly, I conclude that (3) needs to go. And this means that I’ve overestimated the connection between explanation and understanding.

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.

Molinist evolutionary theory

Molinist evolutionary theory (MET) holds that evolutionary theory is correct and based on genuinely random processes. Nonetheless, according to MET, these processes are guided by God. For each random transition (e.g., a random mutation, recombination or selection event) has associated with it a subjunctive conditional of the form "if circumstances C were to occur, then transition T would occur". God non-trivially knows the truth values of all such conditionals, and created the world so as to ensure a sequence of circumstances C that, given the conditionals he knew, would result in a sequence of transitions that fits with his plan.

I have argued elsewhere (a version of this has appeared in Philosophia Christi) that this story undercuts the statistical explanations that evolution needs. Here I want to point out a second issue. We know the probabilities of outcomes of processes in nature essentially by looking at frequencies of outcomes[note 1]. But, almost surely[note 2], a Molinist God can get any sequence of outcomes he wants by tweaking the circumstances appropriately. If a coin is to be flipped a million times, a Molinist God can make them all come out heads not by intervening in the flips, but by ensuring that the conditions C in which the flips happen are such as to make true appropriate conditionals of the form "C→heads".

Given the existence of a Molinist God, one might expect, or so Mike Almeida has argued, observed frequencies that do not match the probabilities involved in the processes. In fact, this might even give rise to an interesting prediction: given a Molinist God, we might expect the more needy to be disproportionately represented among lottery winners, since it seems not unlikely that God would want to choose initial conditions to favor them. If this line of reasoning is right, then given the existence of a Molinist God, the frequencies we observe should not reflect the probabilities of the underlying physical processes. But if so, then our knowledge of the probabilities of the underlying physical processes is undercut. And this is surely a problem for MET, not because it falsifies evolutionary theory, but because it undercuts it epistemically, making it impossible for us to know the probabilistic claims on which evolutionary theory is based.

Suppose, on the other hand, our Molinist rejects the Almeida argument, and holds that even given a Molinist God, the observed frequencies will match the probabilities of the underlying physical processes, perhaps because God would want them to match in order to be a self-effacing creator, or to let us engage correctly in inductive reasoning. In that case, the following is still true. The observed frequencies are not directly evidence for the probabilities of the underlying physical processes. They are only indirectly evidence given some assumptions about how one expects God to act.

Here is another way to put this. On the Molinist view, there is a defeater to our knowledge of probabilities on the basis of frequencies: the frequencies come from God's decision as to the antecedents of conditionals. A controversial thesis about how God chooses to act, if substantiated, would provide a defeater for this defeater. This makes knowledge of probabilities of physical processes rather more roundabout than we think it is. Moreover, I am not clear whether on this view an atheist can know any claims about these probabilities, since God's contingent decision to make the frequencies match the probabilities seems to play a central role.