Theistic Humeanism?

Here’s an option that is underexplored: theistic Humeanism. There are two paths to it.

The path from orthodoxy: Start with a standard theistic concurrentism: whenever we have a creaturely cause C with effect E, E only eventuates because God concurs, i.e., God cooperates with the creaturely causal relation. Now add to this a story about what creaturely causation is. This will be a Humean story—the best I know is the David Lewis one that reduces causation to laws and laws to arrangements of stuff. Keep all the deep theistic metaphysics of divine causation.

The path from heterodoxy: Start with the metaphysics of occasionalism. Don’t change any of the metaphysics. But now add a Humean analysis of creaturely causation in terms of regularities. Since the metaphysics of occasionalism affirms regularities in the world, we haven’t changed the metaphysics of occasionalism, but have redescribed it as actually involving creaturely causation.

The two paths meet in a single view, a theistic Humeanism with the metaphysics of occasionalism and the language of concurrentism, and with creaturely causation described in a Humean way.

This theistic Humeanism is more complex than standard non-theistic Humeanism, but overcomes the central problem with non-theistic Humeanism: the difficulty of finding explanation in nature. If the fact that heat causes boiling is just a statement of regularity, it does not seem that heat explains boiling. But on theistic Humeanism, we have a genuine explanatory link: God makes the water boil because God is aware of the heat.

There is one special objection to theistic Humeanism. It has two causal relations, a divine one and a creaturely one. But the two are very different—they don’t both seem to be kinds of causation. However, on some orthodox concurrentisms, such as Aquinas’s, there isn’t a single kind of thing that divine and creaturely causation are species of. Instead, the two stand in an analogical relationship. Couldn’t the theistic Humean say the same thing? Maybe, though one might also object that Humean creaturely causation is too different from divine causation for the two to count as analogous.

I suppose the main objection to theistic Humeanism is that it feels like a cheat. The creaturely causation seems fake. The metaphysics is that of occasionalism, and there is no creaturely causation there. But if theistic Humeanism is a cheat, then standard non-theistic Humeanism is as well, since they share the same metaphysics of creaturely causation. If non-theistic Humeanism really does have causation, then our theistic Humeanism really does have creaturely causation. If one has fake causation, so does the other. I think both have fake causation. :-)

Humeanism and knowledge of fundamental laws

On a "Humean" Best System Account (BSA) of laws of nature, the fundamental laws are the axioms of the system of laws that best combines brevity and informativeness.

An interesting consequence of this is that, very likely, no amount of advances in physics will
suffice to tell us what the fundamental laws are: significant advances in mathematics will also be needed. For suppose that after a lot of extra physics, propositions formulated in sentences p₁, ..., p_n are the physicist’s best proposal for the fundamental laws. They are simple, informative and fit the empirical data really well.

But we would still need some very serious mathematics. For we would need to know there isn’t a simpler collection of sentences {q₁, ..., q_m} that is logically equivalent to {p₁, ..., p_n} but simpler. To do that would require us to have a method for solving the following type of mathematical problem:

1. Given a sentence s in some formal language, find a simplest sentence s′ that is logically equivalent to s,

in the case of significantly non-trivial sentences s.

We might be able to solve (1) for some very simple sentences. Maybe there is no simpler way of saying that there is only one thing in existence than ∃x∀y(x=y). But it is very plausible that any serious proposal for the laws of physics will be much more complicated than that.

Here is one reason to think that any credible proposal for fundamental laws is going to be pretty complicated. Past experience gives us good reason to think the proposal will involve arithmetical operations on real numbers. Thus, a full statement of the laws will require including a definition of the arithmetical operations as well as of the real numbers. To give a simplest formulation of such laws will, thus, require us to solve the problem of finding a simplest axiomatization of the portions of arithmetic and real analysis that are needed for the laws. While we have multiple axiomatizations, I doubt we are at all close to solving the problem of finding an optimal such axiomatization.

Perhaps the Humean could more modestly hope that we will at least know a part of the fundamental laws—namely the part that doesn’t include the mathematical axiomatization. But I suspect that even this is going to be very difficult, because different arithmetical formulations are apt to need different portions of arithmetic and real analysis.

Humeanism about causation and functionalism about mind

Suppose we combine a Humean account of causation on which causation is a function of the pattern of intrinsically acausal events in reality with a functionalist account of consciousness. (David Lewis, for instance, accepted both.)

Here is an interesting consequence. Whether you are now conscious depends on what will happen in the future. For if the world were to radically change 14 billion years from the Big Bang, i.e., 200 million years from now, in such a way that the regularities that held for the first 14 billion years would not be laws, then the causal connections that require these regularities to be laws would not obtain either, and hence (unless we got lucky and new regularities did the job) our brains would lack the kind of causal interconnections that are required for a functionalist theory of mind.

This dependence of whether we are now conscious on what will happen in the future is intuitively absurd.

But suppose we embrace it. Then if functionalism is the necessary truth about the nature of mind, the fact that we are now conscious necessarily implies that the future will not be such as to disturb the lawlike regularities on which our consciousness is founded. In other words, on the basis of the fact that there are now mental states, one can a priori conclude things about the arrangement of physical objects in the future.

Indeed, this opens up the way for specific reasoning of the following sort. Given what the constitution of humans brains is, and given functionalism, for these brains to exhibit mental states of the sort they do, such-and-such generalizations must be special cases of laws of nature. But for there to be such laws of nature, then the future must be such-and-such. So, we now have a room for substantive a priori predictions of the future.

This all sounds very un-Humean. Indeed, it sounds like a direct contradiction to the Humean idea that reasoning from present to future is merely probabilistic. But while it is very counterintuitive, it is not actually a contradiction to the Humean idea. For on functionalism plus Humeanism about causation, facts about present mental states are not facts about the present—they are facts about the universe as a whole!

(This was sparked by some related ideas by Harrison Jennings.)

A dilemma for best-systems accounts of laws

Here is a dilemma for best-systems accounts of laws.

Either:

1. law-based scientific explanations invoke the lawlike generalization itself as part of the explanation, or

2. they invoke the further fact that this generalization is a law.

Thus, if it is a law that all electrons are charged, and Bob is an electron, on (1) we explain Bob’s charge as follows:

3. All electrons are charged.

4. Bob is an electron.

5. So and that’s why Bob is charged.

But on (2), we replace (3) with:

6. It is a law that all electrons are charged.

Both options provide the Humean with problems.

If it is just the lawlike generalization that explains, then the explanation is fishy. The explanation of why Bob is charged in terms of all electrons being charged seems too close to explaining a proposition by a conjunction that includes it:

7. Bob is charged because Bob is charged and Alice is charged.

Indeed both (3)–(5) and (7) are objectionably cases of explaining the mysterious by the more mysterious: the conjunction is more mysterious than its conjunct and the universal generalization is more mysterious than its instances.

On the other hand, suppose that our explanation of why Bob is charged is that it’s a law that all electrons are charged. This sounds correct in general, but is not appealing on a best-systems view. For on a best-systems view, what the claim that it’s a law that all electrons are charged adds to the claim that all electrons are charged is that the generalization that all electrons are charged is sufficiently informative and brief to make it into the best system. But the fact that it is thus informative and brief does not help it explain anything.

Moreover, if the problem with (3)–(5) was that universal generalizations are too much like conjunctions, the problem will not be relieved by adding more conjuncts to the explanation, namely that the generalization is sufficiently informative and brief.

On two problems for non-Humean accounts of laws

There are three main views of laws:

• Humeanism: Laws are a summing up of the most important patterns in the arrangement of things in spacetime.

• Nomism: Laws are necessary relations between universals.

• Powerism: Laws are grounded in the essential powers of things.

The deficiencies of Humeanism are well known. There are also deficiencies in nomism and powerism, and I want to focus on two.

The first is that they counterintuitively imply that laws are metaphysically necessary. This is well-known.

The second is perhaps less well-known. Nomism and powerism work great for fundamental laws, and for those non-fundamental laws that are logical deductions from the fundamental laws. But there is a category of non-fundamental laws, which I will call impure laws, which are not derivable solely from the fundamental laws, but from the fundamental laws conjoined with certain facts about the arrangement of things in spacetime.

The most notorious of the impure laws is the second law of thermodynamics, that entropy tends to increase. To derive this from the fundamental laws, we need to add some fact about the initial conditions, such as that they have a low entropy. The nomic relations between universals and the essential powers of things do not yield the second law of thermodynamics unless they are combined with facts about which universals are instantiated or which things with which essential powers exist.

A less obvious example of an impure law seems to be conservation of energy. The necessary relations between universals will tell us that in interactions between things with precisely such-and-such universals energy is conserved. And it might well be that the physical things in our world only have these kinds of energy-conserving universals. But things whose universals don’t conserve energy are surely metaphysically possible, and the fact that such things don’t exist is a contingent fact, not grounded in the necessary relations between universals. Similarly, substances with causal powers that do not conserve energy are metaphysically possible, and the non-existence of such things is at best a contingent fact. Thus, to derive the law of conservation of energy, we need not only the fundamental laws grounded in relations between universals or essential powers, but we also need the contingent fact that conservation-violators don’t exist.

Finally, the special sciences (geology, biology, etc.) are surely full of impure laws. Some of them perhaps even merely local ones.

One might bite the bullet and say that the impure laws are not laws at all. But that makes the nomist and powerist accounts inadequate to how “law” gets used in science.

The Humean stands in a different position. If they can account for fundamental laws, impure laws are easy, since the additional grounding is precisely a function of patterns of arrangement. The Humean’s difficulty is with the fundamental laws.

There is a solution, and this is for the nomist and powerist to say that “law of nature” is spoken in many ways, analogically. The primary sense is the fundamental laws that the theories nicely account for. But there are also non-fundamental laws. The pure ones are logical consequences of the fundamental laws, and the impure ones are particularly important consequences of the fundamental laws conjoined with important patterns of things in nature. In other words, impure laws are to be accounted for by a hybrid of the non-Humean theory and the Humean theory.

Now let’s come back to the other difficulty: the necessity worry. I submit that our intuitions about the contingency of laws of nature are much stronger in the case of impure laws than fundamental laws or pure non-fundamental laws. It is not much of a bullet to bite to say that matching charges metaphysically cannot attract—it is quite plausible that this is explained by thevery nature of charge. It is the impure laws where contingency is most obvious: it is metaphysically possible for entropy to decrease (funnily enough, many Humeans deny this, because they define the direction of time in terms of the increase of entropy), and it is metaphysically possible for energy conservation to be violated. But on our hybrid account, the contingency of impure laws is accounted for by the Humean element in them.

Of course, we have to check whether the objections to Humeanism apply to the hybrid theory. Perhaps the most powerful objection to a Humean account of laws is that it only sums up and does not explain. But the hybrid theory can explain, because it doesn’t just sum up—it also cites some fundamental laws. Moreover, it may be the case that the patterns that need to be added to get the impure laws could be initial conditions, such as that the initial entropy is law or that no conservation-violators come into existence. But fundamental law plus initial conditions is a perfectly respectable form of explanation.

On the plurality of bestnesses

According to the best-systems account of laws (BSA), the fundamental laws of nature are the axioms of the system that are true and optimize a balance of informativeness and brevity in a perfectly natural language (i.e., the language cuts reality perfectly at the joints). There are some complications in probabilistic cases, but those will only make my argument below more compelling.

Here is the issue I want to think about: There are many reasonable ways of defining the “balance of informativeness and brevity”.

First, in the case of theories that rule out all but a finite number of worlds, we can say that a theory is more informative if it is compatible with fewer worlds. In such a case, there may be some natural information-theoretic way of measuring informativeness. But in fact, we do not expect the laws of nature to rule out all but a finite number of worlds. We expect them to be compatible with an infinite number of worlds.

Perhaps, though, we get lucky and the laws place restrictions on the determinables in such a way that provides for a natural state space. Then we can try to measure what proportion of that state space is compatible with the laws. This is going to be technically quite difficult. The state space may well turn out to be unbounded and/or infinite dimensional, without a natural volume measure. But even if there is a natural volume measure, it is quite likely that the restrictions placed by the laws make the permitted subset of the state space have zero volume (e.g., if the state space includes inertial and gravitational mass, then laws that say that inertial mass equals gravitational mass will reduce the number of dimensions of the state space, and the reduced space is apt to have zero volume relative to the full space). So we need some way of comparing subsets with zero volume. And mathematically there are many, many tools for this.

Second, brevity is always measured relative to a language. And while the requirement that the language be perfectly natural, i.e., that it cut nature at the joints, rules out some languages, there will be a lot of options remaining. Minimally, we will have a choice point about grouping, Polish notation, dot notation, parentheses, and a slew of other options we haven’t thought of yet, and we will have choice points about the primitive logical operators.

Finally, we have a lot of freedom in how we combine the informativeness and brevity measures. This is especially true since it is unlikely that the informativeness measure is a simple numerical measure, given the zero-volume issue.

We could suppose that there is some objective fact, unknowable to humans, as to what is the right way to define the informativeness and brevity balance, a fact that yields the truth about the laws of nature. This seems implausible. Absent such a fact, what the laws are will be relative to the choice of informativeness and brevity
measure ρ. We might have gotten lucky, and in our world all the measures yield the same laws, but we have little reason to hope for that, and even if this is correct, that’s just our world.

Thus, the story implies that for any reasonable informativeness and brevity measure ρ, we have a concept of a law_ρ. This in itself sounds a bit wrong. It makes the concept of a law not sound objective enough. Moreover, corresponding to each reasonable choice of ρ, it seems we will have a potentially different way to give a scientific explanation, and so the objectivity of scientific explanations is also endangered.

But perhaps worst of all, what BSA had going for it was simplicity: we don’t need any fundamental law or causal concepts, just a Humean mosaic of the distribution of powerless properties. However, the above shows that there is enormous complexity in the account of laws. This is not ideological complexity, but it is great complexity nonetheless. If I am right in my preceding post that at least on probabilistic BSA the fact that something is a law actually enters into explanation, and if I am right in this post that the BSA concept of law has great complexity, then this will end up greatly complicating not just philosophy of science, but scientific explanations.

On probabilistic best-systems accounts, laws aren't propositions

According to the probabilistic best-systems account of laws (PBSA), the fundamental laws of nature are the axioms of the system that optimizes a balance of probabilistic fit to reality, informativeness, and brevity in a perfectly natural language.

But here is a tricky little thing. Probabilistic laws include statements about chances, such as that an event of a certain type E has a chance of 1/3. But on PBSA, chances are themselves defined by PBSA. What it means to say “E has a chance of 1/3” seems to be that the best system entails that E has a chance of 1/3. On its face, this is circular: chance is defined in terms of entailment of chance.

I think there may be a way out of this, but it is to make the fundamental laws be sentences that need not express propositions. Here’s the idea. The fundamental laws are sentences in an formal language (with terms having perfectly natural meanings) and an additional uninterpreted chance operator. There are a bunch of choice-points here: is the chance operator unary (unconditional) or binary (conditional)? is it a function? does it apply to formulas, sentences, event tokens, event types or propositions? For simplicity, I will suppose it’s unary function applying to event types, even though that’s likely not the best solution in the final analysis. We now say that the laws are the sentences provable from the axioms of our best system. These sentences include the uninterpreted chance(x) function. We then say stuff like this:

1. When a sentence that does not use the chance operator is provable from the axioms, that sentence contributes to informativeness, but when that sentence is in fact false, the fit of the whole system becomes  − ∞.

2. When a sentence of the form chance(E) = p is provable from the axioms, then the closeness of the frequency of event type E to p contributes to fit (unless the fit is  − ∞ because of the previous rule), and the statement as such contributes to informativeness.

I have no idea how fit is to be measured when instead of being able to prove things like chance(E) = p, we can prove less precise statements like chance(E) = chance(F) or chance(E) ≥ p. Perhaps we need clauses to cover cases like that, or maybe we can hope that we don’t need to deal with this.

An immediate problem with this approach is that the laws are no longer propositions. We can no longer say that the laws explain, because sentences in a language that is not fully interpreted do not explain. But we can form propositions from the sentences: instead of invoking a law s as itself an explanation, we can invoke as our explanation the second order fact that s is a law, i.e., that s is provable from the axioms of the best system.

This is counterintuitive. The explanation of the evolution of amoebae should not include meta-linguistic facts about a formal language!

Humeans laws and constants

On Mill-Ramsey-Lewis accounts of laws of nature, the laws are the propositions that best balance informativeness and brevity (in a language that cuts nature precisely at the joints).

Now, the laws of nature include constants, such as the fine-structure constant whose current best measured value is 1/137.035999206. Now, we might be lucky, and it might turn out that the fine-structure constant will have some neat and elegant precise value. There is a history of speculation that it has such a value—for a while, there was hope it was exactly 1/137, and then other guesses took over. But suppose we don’t get so lucky. Suppose it just is some messy number with no simple expression. That should, after all, be a serious possibility.

In that case, the exact value of the fine-structure constant cannot be a part of the Mill-Ramsey-Lewis “world in a nutshell” system of laws, since the system would then be infinitely long, and we lose our hope of defining laws in terms of brevity.

So we have two options. First, the system of laws might not include any specific information on the value of the fine-structure constant, but might instead be of the form ∃αF(α) where F(α) says nothing about what α is, except maybe that it’s real-valued and positive. If we go for this option, then we have to say that all the things that depend on the actual value of the fine-structure constant—and that apparently includes all of chemistry—are not in fact laws of nature. This will likely fail to yield some counterfactuals that we want, and while the laws will be briefer, they will be far less informative than if they had something to say about the value of α.

So that moves us to the second option, which is that the laws are of the form ∃αF(α) and F(α) includes some constraints on α, such as that it lies between 1/137.04 and 1/137.03. These constraints are sufficiently tight to generate the nomic implications we need for chemistry and biology. But while this result seems a better fit for science, it is metaphysically very strange. For it is very strange to think that the laws allow the fine-structure constant to have any of an infinite number of values, but these values must lie in a narrow range.

Furthermore, the exact narrow range for α would be determined by fine details (I am not sure if the pun is intended) of exactly how informativeness and brevity are balanced in the definition of the laws.

The same issue comes up for other constants in the laws of nature. Either Mill-Ramsey-Lewis laws do not include anything about the values of constants or else they include oddly specific, but not completely specific, ranges.

Humeans should be (Kenneth-)Pearceans

I have long thought that Humeanism leads to strong inductive scepticism about the future—the thesis that typical inductive generalizations about the future aren’t even more likely than not—roughly because there are a lot more induction-unfriendly worlds with our world’s history than induction-friendly ones.

But this argument assumes that there isn’t some extra-systemic explanation of why we have an induction-friendly physical reality. If there is, then the mere counting of worlds does nothing. Now, standard theism provides such an extra-systemic explanation. But standard theism is incompatible with Humeanism, because God-to-world causation is incompatible with the Humean understanding of causation.

However, it’s occurred to me today that there is a non-standard theism that could furnish the Humean with an escape: Kenneth Pearce has advocated a theism on which God explains the contingent world in a non-causal way.

I don’t know of another option for the Humean in the literature. I know of three candidates for extra-systemic explanations of physical reality:

1. there isn’t one

2. there is one, and it’s theistic

3. there is one, and it’s necessitarian (e.g., Optimalism).

The Humean can’t take the necessitarian way out, because Humeanism is strongly opposed to such necessities. The first option leads to inductive scepticism. That leaves 2. But Humeans cannot accept causal theism. So that leaves them non-causal theism.

Energy conservation

On a Humean metaphysics, energy conservation implies a vast conspiracy in the arrangement of things throughout spacetime, somewhat like this:

1. Wherever there is a change in energy in one region there is a corresponding balancing change in another region.

In an Aristotelian causal powers metaphysics, energy conservation implies a fact like the following about every physical substance x:

2. Every causal power of x whose content includes an effect on the energy of one or more substances also includes a balancing reverse effect on x’s own energy.

That no physical substance simply has a power to affect the energy of another substance, without the content of that power having to include a balancing effect on one’s own energy, is deeply surprising. It is a conspiracy almost as surprising as (1).

These conspiracies strongly suggest that neither the Humean nor the Aristotelian metaphysics is the whole story about energy conservation. The conspiracies desperately call for explanation. I know of two putative explanations: an optimalist one (on which reality strives for value, and mathematically expressible patterns are a part of the value) and a theistic one. Both of these explanations, however, really do great violence to the spirit behind Humean metaphysics. But Aristotelian metaphysics with optimalism or theism explaining the conspiracy in (2) works just fine.

Of course, the problem can also be solved by a different metaphysics, one on which the behavior of objects is explained by pushy global laws. But it is harder to fit human freedom and agency into that metaphysics than into the Aristotelian one.

A problem for some Humeans

Suppose that a lot of otherwise ordinary coins come into existence ex nihilo for no cause at all. Then whether a given coin lies heads or tails up is independent of how all the other coins lie in the sense that no information about the other coins will give you any data about how this one lies.

It is crucial here that the coins came into existence causelessly. If the coins came off an assembly line, and a large sample were all heads-up, we would have good reason to think that the causal process favored that arrangement and hence that the next coin to be examined will also be heads-up.

But now suppose that I know that Humeanism about laws is true, and there is a very, very large number of coins lying in a pile, all of which I know for sure to have come to be there causelessly ex nihilo, and there are no other coins in the universe. Suppose, further, that in fact all the coins happen to lie heads-up. Then when the number of coins is sufficiently large (say, of the order of magnitude of the number of particles in the universe), on Humean grounds it will be a law of nature that coins begin their existence in the heads-up orientation. But if the independence thesis I started the post with is true, then no matter how many coins I examined, I would not have any more reason to think that the next unexamined coin is heads than that it is tails. Thus, in particular, I would not be justified in believing in the heads-up law.

One might worry that I couldn’t know, much less know for sure, that the coins are there causelessly ex nihilo. A reasonable inference from the fact that lots of examined coins are all heads-up would seem to be that they were thus arranged by something or someone. And if I made that inference, then I could reasonably conclude that the coins are all heads-up. But my conclusion, while true and justified, would not be knowledge. I would be in a Gettier situation. My justification depends essentially on the false claim that the coins were arranged by something or someone. So even if one drops the assumption that I know that the coins are there causelessly ex nihilo, I still don’t know that the heads-up law holds. Moreover, my reason for not knowing this has nothing to do with dubious theses about the infallibility of knowledge. I don’t know that the heads-up law holds, whether fallibly or infallibly.

There is no problem for the Humean as yet. After all, there is nothing absurd about there being hypothetical situations where there is a law but we can’t know that it obtains. But for any Humean who additionally thinks that our universe came into existence causelessly, there is a real challenge to explain why the laws of our world are not like the heads-up law—laws that we cannot know from a mere sample of data.

This problem is fatal, I think, to the Humean who thinks that our universe started its existence with a large number of particles. For the properties of the particles would be like the heads-up and tails-up orientations of the coins, and we would not be in a position to know all particles fall into some small number of types (as the standard model in particle physics does). But a Humean scientist who doesn’t think the universe has a cause could also think that our universe started its existence with a fairly simple state, say a single super-particle, and this simple state caused all the multitude of particles we observe. In that case, the order-in-multiplicity that we observe would not be causeless, and the above argument would not apply.

Humean metaphysics implies Cartesian epistemology

Let’s assume two theses:

1. Humean view of causation.

2. Mental causalism: mental activity requires some mental states to stand in causal relations.

If I accept these two theses, then I can a priori and with certainty infer a modest uniformity of nature thesis. Here’s why. On mental causalism, mental activity requires causation. On Humeanism, causation depends on the actual arrangement of matter. If the regularities found in my immediate vicinity do not extend to the universe as a whole, then they are no causal laws or causal relations. Thus, given causalism and Humeanism, I can infer a priori and with certainty from the obvious fact that I have mental states that there are regularities in the stuff that my mind is made of that extend universally. In other words, we get a Cartesian-type epistemological conclusion: I think, so there must be regularity.

In other words, Humean metaphysics of nature plus a causalist theory of mind implies a radically non-Humean epistemology of nature. The most plausible naturalist theories of mind all accept causalism. So, it seems, that a Humean metaphysics of nature plus naturalism—which is typically a part of contemporary Humean metaphysics—implies a radically non-Humean epistemology of nature.

So Humean metaphysics and epistemology don’t go together. So what? Why not just accept the metaphysics and reject the epistemology? The reason this is not acceptable is that the Cartesian thesis that the regularity of nature follows with certainty from what I know about myself is only plausible (if even then!) given Descartes’ theism.

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.