Wednesday, May 15, 2013

Random numbers and their sequences

Bear with a simple and standard bit of mathematics: the mathematics may give us lessons about God and evolution, frequentism, single-case chances and Humean views of causation.

Consider the following standard one-to-one and onto map between the interval [0,1] and the space [0,1]ω of infinite sequences of numbers from that interval. The map starts with a single decimal number x=0.d1d2d3... in [0,1][note 1] and generates an infinite sequence ψ(x)=(ψ1(x),ψ2(x),...) by taking every second digit of x after the decimal point and letting that define ψ1(x), then discarding these digits, taking every second digit of what remains and letting that define ψ2(x), and so on. Thus, ψ1(x)=0.d1d3d5..., ψ2(x)=0.d2d6d10..., ψ3(x)=0.d4d12d20..., and so on.

Interestingly, ψ not only shows that [0,1] and [0,1]ω have the same cardinality, but if we equip [0,1] with a uniform probability measure and [0,1]ω with an infinite product of uniform probability measures, i.e., let [0,1]ω be the probability space modeling infinite independent choices of uniformly distributed numbers in [0,1], then it turns out that ψ is a probability-preserving isomorphism. Hence, the two probability spaces are probabilistically isomorphic. There is, thus, "nothing more" to choosing an infinite sequence of uniformly distributed numbers in [0,1] than there is to choosing a single such number.

And of course what goes for [0,1] and [0,1]ω also goes for finite sequences: the probability-preserving isomorphism between [0,1] and [0,1]n is even easier to construct.

There are some potential philosophical consequences of this isomorphism: it shows that there is no principled difference between single-case and sequences, when we're willing to deal with continuous outcomes (there is when we have a finite outcome space).

Lesson 1: Anybody who believes in the utter impossibility of single-case chances or probabilities, including for continuous-valued events like decay times or darts thrown at boards, should believe in the utter impossibility of chances or probabilities in the case of infinite sequences as well.

Thus, Lesson 2: Frequentism is dubious.

Lesson 3: If probabilistic causation with continuous-valued outcomes is possible, single-case probabilistic causation should be possible, and in particular single-case causation should be possible. For there is in principle no difference between single-case and sequential probabilities.

Thus, Lesson 4: Humeanism about causation is dubious.

Lesson 5: Given that it is plausible that if God intentionally and specifically chooses just a single real number in [0,1] with full precision, that real number isn't genuinely random in the sense scientists like biologists or quantum physicists mean, neither will an infinite sequence of divine choices embody randomness. Hence, reconciliations between random evolution and exhaustive divine planning of every particular event fail.

Heath White said...

Re: Lesson 5: I've always taken the "random" in "random evolutionary process" to mean something like "unspecifiable by any theory framed at the level of description we're dealing with." That is, if Newtonian determinism had turned out to be true, I don't think biologists would have had to give up on "random" evolutionary processes. Their point, presumably, would be that full biological descriptions of species and environments would not give you enough information to predict the course of species development. Along the same lines, gametes in sexual reproduction merge "randomly" in the sense that a complete genetic description of Mom's DNA and Dad's DNA will not tell you what Junior's DNA will look like.

Any stronger notion of randomness involves biologists giving chemists and physicists hostages to fortune, yes?

And by the same logic, there is no conflict between this kind of randomness and divine control of outcomes, either.

Alexander R Pruss said...

Well, the randomness here has to be of a sort that can be plugged into statistical explanations.

Deterministic physics MIGHT by itself be OK here, because we can then have the sort of randomness given by the principle of indifference as applied to the initial conditions of the universe. But if God is choosing specific outcomes, whether on a case-by-case basis in the indeterministic Thomistic story or by setting the initial conditions in the deterministic story, there is no indifference any more.

(That said, explanation by indifference is something I am not confident about, since the principle of indifference is dubious.)

Crude said...

Alex,

Given that it is plausible that if God intentionally and specifically chooses just a single real number in [0,1] with full precision, that real number isn't genuinely random in the sense scientists like biologists or quantum physicists mean, neither will an infinite sequence of divine choices embody randomness. Hence, reconciliations between random evolution and exhaustive divine planning of every particular event fail.

I recall Stephen Barr talking about this - but his claim was that what scientists mean when they talk about randomness (and what they can only mean, for their explanations to truly be scientific) necessarily falls short of a claim about there being no guidance, no purpose, no direction at work, period.

This doesn't seem to be a view specific to Barr. Eugenie Scott seemed to concede as much during the NABT controversy, Sober seems to argue as much, etc.

When you say "that real number isn't genuinely random in the sense scientists like biologists or quantum physicists mean", that seems to suggest biologists and quantum physicists, as biologists and quantum physicists (rather than as, say, naturalists or something else), are talking about what God does and doesn't do in nature. But that seems absurd. How would such a claim ever be tested in science? Especially if science is restrained by methdological naturalism?

Alexander R Pruss said...

1. Imagine that in the history of the universe is only ever one physically undetermined event other than the Big Bang:
David throws a perfectly round stone at a large perfectly round target while blindfolded. And God, to express his love for David, makes the stone hit the exact geometrical center of the target.

This story is coherent. But what sense could there be in adding that the stone hit the center "at random" or that it had equal objective "chance" of hitting every other place on the target, unless we suppose an unlikely hypothesis that God decided at random, with equal chance for every point, where to make the stone impact?

So if there were only ever one physically undetermined event, and it were fully determined by God, there would be no sense in saying that that event was "random" or talking of objective "chances" of the event turning out this way or that.

And then by the argument in the post, I note that a single event is probabilistically isomorphic to an infinite sequence of events, so the same goes for an infinite sequence.

2. "How would such a claim ever be tested in science?" Take my David story. We have two hypotheses. On H1, the stone hits a random point on the target according to some continuous distribution (maybe a Gaussian centered on the center of the target). On H2, God decides where the stone hits. Let E be the evidence, namely that the stone hit the exact center. (We can approximate, of course, if we can't measure the exactness.) The probability of E on H1 is zero (or maybe infinitesimal--but I have a bunch of articles arguing against that). The probability of E on H2 may be small. After all, there are many things God could choose to do with David here: he could make David humbler by making the stone hit completely off-target, etc. But nonetheless a reasonable thing for God to do is to make the stone hit the exact center in order to manifest his love for David. Thus, plausibly, the probability of E on H2 is greater than zero (and more than merely infinitesimal). So, P(E|H1)=0 and P(E|H2)>0. Therefore, E is very strong evidence for H2 over H1.

On the other hand, suppose that the stone hits a point that does not appear to be at all "special" in a way that a perfect being should care about. That would (very) weakly confirm H1 over H2.

3. "Especially if science is restrained by methdological naturalism?" If methodological naturalism is to be compatible with scientific realism, it had better be compatible with ruling out at least some supernaturalistic hypotheses. For instance, our evidence for the germ theory of disease is evidence against a direct demonic intervention theory of disease.

4. Yes, a lot of people like Barr do say things like that. But I am afraid that they have too weak a view of how stochastic explanation works.

5. Here's a simple argument to show that there is a problem. Suppose that I toss an indeterministic coin, and according to physics we learn that the probability that the coin lands is heads. Moreover, suppose that the right theology tells us that necessarily God determines how coins land. Then: P(God chooses to have the coin land heads) = P(the coin lands heads) = 1/2. But how could we be able to assign exact numerical probabilities to divine choices?

Crude said...

Alex,

So if there were only ever one physically undetermined event, and it were fully determined by God, there would be no sense in saying that that event was "random" or talking of objective "chances" of the event turning out this way or that.

I think the point to take away from talk like this may be that it's outside of a reasonable scope of science to claim that anything happens randomly, in the strong sense you're using here.

Now, there's looser standards of random that could be used. Say, random with respect to X ('The position of the sun has no effect on the outcome of process A'), or at least a qualified standards ('Based on what we're able to discern, the position of the sun has no effect on the outcome of process A').

On the other hand, suppose that the stone hits a point that does not appear to be at all "special" in a way that a perfect being should care about.

I don't know how we'd go about determining what a perfect being should or shouldn't care about, at all. Now, I think - outside of science - we may have various intuitions or ideas that may make one result or another more plausible. But the accent there should be on the 'outside of science'. Inside of science, I think it would be downright bizarre to have evolution presented as a scientific theory which hinges on a positive claim about how God does or does not act. Not if it wants to be a scientific rather than a metaphysical view.

If methodological naturalism is to be compatible with scientific realism, it had better be compatible with ruling out at least some supernaturalistic hypotheses. For instance, our evidence for the germ theory of disease is evidence against a direct demonic intervention theory of disease.

In what sense - total? Partial? The existence of germ theory wouldn't seem to be evidence that demonic intervention never takes place. (Of course I'd add, I'm not exactly certain how to rule 'demons' as supernatural or natural, except in the loosest and least helpful senses.)

On the other hand, maybe MN is only partly compatible with scientific realism when all is said and done, proceeding on a case by case basis.

Here's a simple argument to show that there is a problem. Suppose that I toss an indeterministic coin, and according to physics we learn that the probability that the coin lands is heads. Moreover, suppose that the right theology tells us that necessarily God determines how coins land. Then: P(God chooses to have the coin land heads) = P(the coin lands heads) = 1/2. But how could we be able to assign exact numerical probabilities to divine choices?

I'm having trouble seeing where the problem is coming in. Classical theism maintains that God maintains the existence of the universe from moment to moment. I suppose you could phrase this as 'God chooses to maintain the existence of the universe from moment to moment'. He's apparently been choosing 'Yes, I will maintain' for quite a while now. Is there something wrong with God's 'decisions' on this front being absurdly predictable?

Alexander R Pruss said...

1. "there's looser standards of random that could be used. Say, random with respect to X ('The position of the sun has no effect on the outcome of process A'), or at least a qualified standards ('Based on what we're able to discern, the position of the sun has no effect on the outcome of process A')"

But notice that the randomness has to be of a sort that can be used in explanations. We want to be able to say things like: "The coin tossed a hundred times landed heads approximately 50 times because it was very likely to." But for the "because it was very likely to" part to be true, we need more than just the irrelevance of the position of the sun, etc. Such irrelevance claims are negative. But the explanation is something positive. Chance plays a positive role in this explanation.

Here's where I think there may be some talking-past. It's not so much that I start with a strong view of chance. Rather, I start with a strong view of explanation. Correct explanations tell us about objective connections in the world. They do not simply organize the data into patterns, but remove mystery, telling us why things happened as they did. Correct explanations never disappear with more information--if an apparent explanation disappears with more information, that's because it wasn't correct.

Now, I think the kinds of "chancy explanations" you, Barr and others may be thinking about do disappear with more information, such as information about divine intentions.

2. "I don't know how we'd go about determining what a perfect being should or shouldn't care about, at all." Here's a simple strategy. We first determine what is good, what is bad and what is indifferent. Presumably, we can do that to some extent, or else moral scepticism follows. Well, we now know that the perfect being desires to promote the good things, and to do so in proportion to how good they are, to impede the bad things, and to do so in proportion to how bad they are, and doesn't care about the indifferent ones (if there are any). For that just follows from the concept of moral perfection.

3. "The existence of germ theory wouldn't seem to be evidence that demonic intervention never takes place."

Sure. But the demonic theory I had in mind was one that would say that all or most instances of disease are caused by direct demonic action.

4. I like the point about maintenance of the universe. So, yes, we can say: "It's likely that things that exist now will continue to exist later." We can say this because such predictability and maintenance is valuable, and hence is what we expect from a perfect being.

Crude said...

Alex,

Here's where I think there may be some talking-past. It's not so much that I start with a strong view of chance. Rather, I start with a strong view of explanation. Correct explanations tell us about objective connections in the world. They do not simply organize the data into patterns, but remove mystery, telling us why things happened as they did. Correct explanations never disappear with more information--if an apparent explanation disappears with more information, that's because it wasn't correct.

I think when an theory requires what amounts to grand metaphysical/philosophical and theological assumptions in order to really 'work' (in this case, an assumption that there is no omnicient, omnipotent God, or even no beings of sufficient knowledge and power), it's a good sign you're now dealing with something that is not a scientific project. So if it comes down to the wire such that the only way to interpret evolutionary theory is in terms of claims about the inaction or lack of knowledge of God, etc, that's the point where I'm forced to say evolutionary theory isn't scientific after all.

In truth, I don't think that's nearly the case. Instead, it seems that scientific theories have a limit on their granularity, so to speak - we can talk about laws and habits and so on and how all these things work out. We can have models and idealizations and so on, and all of this can be useful. But, these are forever qualified and limited theories and models.

Now, I think the kinds of "chancy explanations" you, Barr and others may be thinking about do disappear with more information, such as information about divine intentions.

I'm not sure about this. Statements of chance (putting aside quantum stuff for a moment) are statements of ignorance. Go back to your coin flip example. Let's say we live in a Laplacean universe. Does that mean all odds calculations are pointless? After all, any given real-world odds calculation would be incorrect - no, there's not a 57% chance of X happening. It either will happen or it will not. But it doesn't seem right to therefore regard odds models that rely on odds calculations as unscientific, especially if they're the best we have.

Here's a simple strategy. We first determine what is good, what is bad and what is indifferent. Presumably, we can do that to some extent, or else moral scepticism follows. Well, we now know that the perfect being desires to promote the good things, and to do so in proportion to how good they are, to impede the bad things, and to do so in proportion to how bad they are, and doesn't care about the indifferent ones (if there are any).

I have a few concerns here. First, while a Catholic like myself is concerned with God (and thus I'm dealing with omnipotence, etc), all bets seem to be off when talking about any of the other wide variety of non-Judaeo-Christian God/gods out there. Second, knowing that God will promote the good and demote the bad helps in a broad sense, but there seem to be a whole lot of ways to do either - so how do I know which one God chose to use? Third, with the 'indifferent', I'm not sure indifference is a guarantee of non-action. Certainly it wouldn't be a guarantee of non-awareness.

I like the point about maintenance of the universe. So, yes, we can say: "It's likely that things that exist now will continue to exist later." We can say this because such predictability and maintenance is valuable, and hence is what we expect from a perfect being.

Alright. But then that seems to remove worry about 'assigning odds to God's actions' based on our study of nature, given all the appropriate caveats. It turns out we can assign reasonably some odds to some divine decisions. In other cases, they get more blurry, maybe even practically impossible to make.

Neil Bates said...

Alexander, doesn't your argument about particular compared to infinite cases, support (broadly suggest at least) that the infinite cases aren't as paradoxical as you have thought?