Causal finitism/countabilism and 4D unrestricted composition

Assume four dimensional unrestricted composition (or mereological universalism). Plausibly, there is a world where on Monday there are infinitely many things, and on Tuesday Alice and Bob cause a sofa to go into the living room.

Also plausibly:

1. If x and y cause z at t, then anything composed of x and y at t causes z at t.

Why? Well, anything composed of x and y at t has the causal powers of x and y at t, and these causal powers are activated, and their activation results in z, so it seems that anything so composed causes z.

But on Monday there were infinitely many things. Thus, by four dimensional unrestricted composition, there are infinitely many things composed on Monday of some of the infinitely many things that exist on Monday and composed on Tuesday of Alice and Bob. By (1), each of these things is the cause of the sofa’s entry into the living room.

Thus, infinitely many things are the cause of the sofa’s entry into the living room. This violates causal finitism. Hence we have an argument from causal finitism against 4D unrestricted composition.

In fact, we have an argument from causal countabilism against 4D unrestricted composition, since there are uncountably many things composed on Monday of the infinitely many things that exist then and composed on Tuesday of Alice and Bob.

Causal countabilism

Causal finitism says that every item has a finite causal history.

A weaker view is causal countabilism, that all the causal histories of items are at most countably infinite.

Is there any evidence for causal countabilism? Well, of course, any argument for causal finitism is an argument for causal countabilism. But is there any other evidence?

I can think of one piece. It would be good if we could come up with a metaphysical account of the countable, maybe to resolve the Lowenheim-Skolem theorem, or to help provide a metaphysical characterization of the natural numbers. A causal countabilism that isn’t a causal finitism could do that: a set is countable if and only if it is the same size as a possible causal history of an event. If a theory can help accomplish an important task, maybe that’s some evidence for the theory. (That said, causal finitism can also account for countability, as I discuss in my infinity book.)

Are there any known applications for causal countabilism? Before today, I would have said “no”. But I now know two. One is that the “technical assumption” in my last version of the Meyer cosmological argument is entailed by causal countabilism. A second will be in my next post today.

Explanatory infinitism

I wish one could formulate and defend a version of explanatory finitism, roughly saying that nothing has infinitely many explanations or an infinite chain of explanations, even if these explanations are not causal (causal finitism of course rules out the causal case). But it seems hard to do so. There do seem to be counterexamples to explanatory finitism. Here some I’m thinking about right now:

1. Why are there infinitely many primes? Because there are infinitely many primes bigger than two. Why are there infinitely many primes bigger than two? Because there are infinitely many primes bigger than three. And so on.

2. Alice has just gone to heaven. Why is she going to experience a day of bliss? Because she will have a day of bliss tomorrow. And also because she will have a day of bliss the day after tomorrow. And also because she will have a day of bliss the day after that. And so on.

3. Alice has just gone to heaven. Why is she going to experience bliss on every future day? Partly because she will have bliss tomorrow. And partly because she will have bliss the day after tomorrow. And so on.

An Aristotelian argument for Causal Finitism

The Aristotelian account of individuation for beings in the same species proceeds by matter. But to fill out this account, we may need to talk about the times at which the matter is different.

One reason for talking about this is the possibility on some versions of Aristotelianism that a chunk of matter can survive substantial change, and hence can compose different substances—albeit at different times. If one wolf kills and eats another, there is in principle no reason why eventually the cannibal wolf’s matter might not entirely be the same as the matter of the victim, then.

So, perhaps, we may want to say that what distinguishes members x and y of the same material species is that they have different matter at the same time. But that’s not quite right. For it could be that x and y don’t ever exist at the same time! We might thus want to say that what grounds the distinction between x and y is:

1. at some time t, either x exists and y doesn’t or x doesn’t exist and y does or x and y both exist and have different matter.

This has at least two problems. First, if I travel backwards in time to a time when I already existed, then I will have different matter from my past self at the same time (since I’ve changed some cells in the meanwhile).

But the second is probably the more serious. According to the story in (1), past, present and future differences in matter all ground the distinction between x and y. But it is implausible that future differences in matter be relevant to my present distinction from you. So we should minimally restrict (1) to talking of past and present times t. But if we do that, then what grounds our difference changes over time, and does so pointlessly, because as soon as our distinction has once been established, that’s surely all we need. In fact, the last point is particularly important given the soul’s survival past death: what makes your soul past death different from mine is presumably the past difference in matter. This suggests that if we want to get at what truly grounds our distinction, we have to go back in time as far as we can, to the very beginnings of our existence, and say:

2. either x and y differ in the time at which they began to exist, or x and y began at the same time with different matter.

There is another reason to prefer (2) to (1). We not only want an account of the distinction of substances within a world, but between possible worlds. But there are many possible worlds where I exist but where my current matter is different from the matter I now have in the actual world (namely, because my diet was different). Thus, (1) has no hope of being extended to an account of transworld distinction. But (2) has some hope of such an extension. At the very least, it gives a sufficient condition for distinctness in the transworld case, while (1) does not even do that.

But now here is an interesting thing. If (2) is the true story about the distinction between material substances of the same species, it follows that:

3. It is impossible for there to be two material substances of the same species that have no beginning in time.

But if Causal Finitism (the doctrine that one effect cannot have infinitely many causes) is false, it seems like it should be possible to have an organism that has lived for an infinite amount of time, and likewise possible to have two such organisms in one species. So (2) gives us a new reason to accept Causal Finitism.

However, personally, I don’t like the Aristotelian account of distinctions between substances.

A theological argument against Causal Finitism

Some people are attracted to this idea:

1. God gives infinitely many chances for everyone to be saved.

At the same time, by and large (with the main exception being Origen) the Christian tradition accepts:

2. Whether you are saved or not is determined at the time of death.

It’s interesting to note that if you deny Causal Finitism, you can have both (1) and (2). You can suppose, for instance, that God miraculously sets up a supertask for you in the millisecond prior to death, miraculously accelerating the functioning of your soul in such a way that in the first half millisecond you have one choice whether to be saved or not, in the next quarter millisecond you have another choice, in the next eighth you have another, and so on. And then you’re saved if any of these infinitely many choices was made for God, and damned only if all them were against God.

While I buy Causal Finitism, and hence probably have to deny the conjunction of (1) and (2) since on (1) and (2) your state post-death appears to be determined by infinitely many causes (the choices), I think this is a somewhat attractive reason to deny Causal Finitism.

From chickens and eggs to a cosmological argument

Suppose we have an infinite regress of chickens and eggs, with no first item, and suppose that if there is any last item, it’s not a chicken.

Plausibly, then:

1. The plurality of the chickens explains the eggs.

But also plausibly:

2. If x has an explanation, then x has a fundamental explanation.

However:

3. No plurality of chickens and eggs fundamentally explains the eggs.

4. So, something that is other than a plurality of chickens and eggs fundamentally explains the eggs.

Why is (3) true? Well, there are two kinds of chicken and egg pluralities.

1. Pluralities with an earliest item.
2. Pluralities with no earliest item.

A Type I plurality does not explain the chickens and eggs, because it does not explain any chickens and eggs earlier than the earliest item in the plurality.

A Type II plurality at best non-fundamentally explains the eggs. For given a Type II plurality, choose any item x in the plurality. Item x is a chicken or egg, and there is an earlier chicken or egg y in the plurality. Since y explains x, by a plausible case of transitivity, the subplurality where we drop x also explains the eggs. So the initial Type II plurality did not fundamentally explain the eggs.

Since every plurality of chickens and eggs is of Type I or Type II, we have (3).

The argument above applies beyond chickens and eggs. Suppose:

5. Every physical item in the world is caused.

Then:

6. Either there is a physical item with a non-physical cause or every physical item has a physical cause.

Add:

7. There are no loops of physical causes.

8. Physical causation is transitive.

As I show in my 1999 paper using Zorn’s Lemma, if every physical item has a physical cause, it is then possible to partition the physical items into an “even” and an “odd” subset, where every item in the even subset causes an item in the odd subset, and every item in the odd subset causes an item in the even subset. Call the odd items “eggs”. Call the non-final even items “chickens” (a non-final item is one that causes a further physical item). Then, plausibly, the chickens explain the eggs. By the argument above, something non-physical thus explains the eggs. Anything non-physical that explains the eggs explains all of physical reality (because for any physical item, it is either an egg or there is an egg prior to it). So:

9. There is a non-physical item that explains all of physical reality.

I think the thing that needs the most defense is premise (2). It’s somewhat close to causal finitism.

A long walk

Alice has lived forever in a universe with an infinite road that has a beginning and no end, and is marked every mile. Every day of her life, by an irresistable longing, she has followed these rules:

1. If somehow she’s not on the road, she goes to mile zero on the road.

2. If she is on the road, she walks a mile in the endless direction of the road.

3. Besides the movement required by 1 and 2, she stays in place.

Where is Alice now? Nowhere! There is no possible present scenario compatible with the rules. She can’t either be at mile zero or off the road, because if two days ago she was on the road, she would now be on the road now be past mile zero, and if two days ago she wasn’t on the road, she would now be on the road past mile zero. She can’t be at mile n, because then she would have to have been off the road some time back, which violates the previous argument.

The rules are all coherent. A person who has to follow these rules seems to be possible. Yet the story is impossible. What went wrong? The neatest explanation seems to be that a (causally connected) infinite past is impossible.

(This is inspired by a recent infinite past Grim Reaper story I read from Rob Koons.)

Classical mereology and causal regresses

Assume classical mereology with arbitrary fusions.

Further assume two plausible theses:

1. If each of the ys is caused by at least one of the xs and there is no overlap between any of the xs and ys, then the fusion of the ys is caused by a part of the fusion of the xs.

2. It is impossible to have non-overlapping objects A and B such that A is caused by a part of B and B is caused by a part of A.

It follows that:

3. It is impossible to have an infinite causal regress of non-overlapping items.

For suppose that A₀ is caused by A₋₁ which is caused by A₋₂ and so on. Let E be a fusion of the even-numbered items and O a fusion of the odd-numbered ones. Then by (1), a part of E causes O and a part of O causes E, contrary to (2).

This is rather like explanatory circularity arguments I have used in the past against regresses, but it uses causation and mereology instead.

Another infinite dice game

Suppose infinitely many people independently roll a fair die. Before they get to see the result, they will need to guess whether the die shows a six or a non-six. If they guess right, they get a cookie; if they guess wrong, an electric shock.

But here’s another part of the story. An angel has considered all possible sequences of fair die outcomes for the infinitely many people, and defined the equivalence relation ∼ on the sequences, where α ∼ β if and only if the sequences α and β differ in at most finitely many places. Furthermore, the angel has chosen a set T that contains exactly one sequence from each ∼-equivalence class. Before anybody guesses, the angel is going to look at everyone’s dice and announce the unique member α of T that is ∼-equivalent to the actual die rolls.

Consider two strategies:

1. Ignore what the angel says and say “not six” regardless.

2. Guess in accordance with the unique member α: if α says you have six, you guess “six”, and otherwise you guess “not six”.

When the two strategies disagree for a person, there is a good argument that the person should go with strategy (1). For without the information from the angel, the person should go with strategy (1). But the information received from the angel is irrelevant to each individual x, because which ∼-equivalence class the actual sequence of rolls falls into depends only on rolls other than x’s. And following strategy (1) in repeats of the game results in one getting a cookie five out of six times on average.

However, if everyone follows strategy (2), then it is guaranteed that in each game only finitely many people get a shock and everyone else gets a cookie.

This seems to be an interesting case where self-interest gets everyone to go for strategy (1), but everyone going for strategy (2) is better for the common good. There are, of course, many such games, such as Tragedy of the Commons or the Prisoner’s Dilemma, but what is weird about the present game is that there is no interaction between the players—each one’s payoff is independent of what any of the other players do.

(This is a variant of a game in my infinity book, but the difference is that the game in my infinity book only worked assuming a certain rare event happened, while this game works more generally.)

My official line on games like this is that their paradoxicality is evidence for causal finitism, which thesis rules them out.

A dialectically failing argument for truth-value realism about arithmetic

Truth-value realism about (first-order) arithmetic is the thesis that for any first-order logic sentence in the language of arithmetic (i.e., using the successor, addition and multiplication functions along with the name “0”), there is a definite truth value, either true or false.

Now, consider the following argument for truth-value realism about arithmetic.

Assume eternalism.

Imagine a world with an infinite space and infinite future that contains an ever-growing list of mathematical equations.

At the beginning the equation “S0 = 1” is written down.

Then a machine begins an endless cycle of alternation between three operations:

1. Scan the equations already written down, and find the smallest numeral n that occurs in the list but does not occur in an equation that starts with “Sn=”. Then add to the bottom of the list the equation “Sn = m” where m is the numeral coming after n.

2. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n + m= does not occur in the list of equations, and write at the bottom of the list n + m = r where r is the numeral representing the sum of the numbers represented by n and m.

3. Scan the equations already written down, and find the smallest pair of numerals n, m (ordered lexicographically) such that n ⋅ m= does not occur in the list of equations, and write at the bottom of the list n ⋅ m = r where r is the numeral representing the product of the numbers represented by n and m.

No other numerals are ever written down in that world, and no equations disappear from the list. We assume that all tokens of a given numeral count as “alike” and no tokens of different numerals count as “alike”. The procedure of producing numerals representing sums and products of numbers represented by numerals can be given entirely mechanically.

Now, if ϕ is an arithmetical sentence, then we say that ϕ is true provided that ϕ would be true in a world such as above under the following interpretation of its basic terms:

4. The domain consists of the first occuring token numerals in the giant list of equations (i.e., a token numeral in the list of equations is in the domain if and only if no token alike to it occurs earlier in the list).

5. 0 refers to the zero token in the first equation.

6. The value of Sn for a token numeral n is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a capital S token followed by a token alike to n.

7. The value of n + m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a plus sign followed by a token alike to m.

8. The value of n ⋅ m for token numerals n and m is the token in the domain alike to a token appearing after the equal sign in an equation whose left-side consists of a token alike to n follow by a multiplication sign followed by a token alike to m.

It seems we now have well-defined truth-value assignments to all arithmetical sentences. Moreover, it is plausible that these assignments would be correct and hence truth-value realism about arithmetic is correct.

But there is one serious hole in this argument. What if there are two worlds w₁ and w₂ with lists of equations both of which satisfy my description above, but ϕ gets different truth values in them? This is difficult to wrap one’s mind around initially, but we can make the worry concrete as follows: What if the two worlds have different lengths of “infinite future”, so that if we were to line up the lists of equations of the two worlds, with equal heights of lines, one of the two lists would have an equation that comes after all of the equations of the other list?

This may seem an absurd worry. But it’s not. What I’ve just said in the worry can be coherently mathematically described (just take a non-standard model of arithmetic and imagine the equations in one of the lists to have the order-type of that model).

We need a way to rule out such a hypothesis. To do that, what we need is a privileged notion of the finite, so that we can specify that for each equation in the list there is only a finite number of equations before it, or (equivalently) that for each operation of the list-making machine, there are only finitely many operations.

I think there are two options here: a notion of the finite based on the arrangement of stuff in our universe and a metaphysically privileged notion of the finite.

There are multiple ways to try to realize the first option. For instance, we might say that a finite sequence is one that would fit in the future of our universe with each item in the sequence being realized on a different day and there being a day that comes after the whole sequence. (Or, less attractively, we can try to use space.) One may worry about having to make an empirical presupposition that the universe’s future is infinite, but perhaps this isn’t so bad (and we have some scientific reason for it). Or, more directly in the context of the above argument, we can suppose that the list-making machine functions in a universe whose future is like our world’s future.

But I think this option only yields what one might call “realism lite”. For all we’ve said, there is a possible world whose future days have the order structure of a non-standard model of arithmetic, and the analogue to the mathematicians of our world who employed the same approach as we just did to fix the notion of the finite end up with a different, “more expansive”, notion of the finite, and a different arithmetic. Thus while we can rigidify our universe’s “finite” and or the length of our universe’s future and use that to fix arithmetic, there is nothing privileged about this, except in relation to the actual world. We have simply rigidified the contingent, and the necessity of arithmetical truths is just like the necessity of “Water is H₂O”—the denial is metaphysically impossible but conceivable in the two-dimensionalist sematics sense. And I feel that better than this is needed for arithmetic.

So, I think we need a metaphysically privileged notion of the finite to make the above argument go. Various finitism provide such a notion. For instance, finitism simpliciter (necessarily, there are only finitely many things), finitism about the past (necessarily, there are always only finitely many past items), causal finitism (necessarily, each item has only finitely many causal antecedents), and compositional finitism (necessarily, each item has at most finitely many parts). Finitism simpliciter, while giving a notion of the finite, doesn’t work with my argument, since my argument requires eternalism, an infinite future and an ever-growing list. Finitism about the past is an option, though it has the disadvantage that it requires time to be discrete.

I think causal finitism is the best option for what to plug into the argument, but even if it’s the best option, it’s not a dialectically good option, because it’s more controversial than the truth-value realism about arithmetic that is the conclusion of the argument.

Alas.

Probabilities of regresses of chickens

Suppose we have a backwards-infinite sequence of asexually reproducing chickens, ..., c₋₃, c₋₂, c₋₁, c₀ with c_n having a chance p_n of producing a new chicken c_n + 1 (chicken c₀ may or may not have succeeded; the earlier ones have succeeded). Suppose that the p_n are all strictly between 0 and 1, and that the infinite product p₋₁p₋₂p₋₃... equals some number p strictly between 0 and 1.

Intuitively, we should be surprised that chicken c₀ exists if p is low and not surprised if p is high. If we have observed c₀ and are considering theories as to what the chances p_n are, other things being equal, we should prefer the theories on which the product p is high to ones on which it’s low.

But what exactly does p measure? It seems to be some kind of a chance of us getting c₀. But it doesn’t measure the unconditional probability of getting an infinite sequence of chickens leading up to c₀. For that is very tiny indeed, since it is extremely unlikely that the world would contain chickens at all. It seems to be a kind of conditional probability. Let q_n be the proposition that chicken c_n exists. Then P(q₀∣q_n) = p₀p₋₁p₋₂...p_n, and so p is the limit of the conditional probabilities P(q₀∣q_n). It is plausible thus to think of p as a conditional probability of q₀ on q_−∞, which is the infinite disjunction of all the q_n.

But q_−∞ is a rather odd proposition. It is grounded in q_n for every finite n, assuming that a disjunction, even an infinite one, is grounded in its true disjuncts. Thus every one of the q_n is explanatorily prior to q_−∞. But this means that P(q₀∣q_−∞) is actually a conditional probability of q₀ on something that isn’t explanatorily prior to q₀—indeed, that is explanatorily posterior to q₀. This challenges the interpretation of p as a chance of getting chicken c₀.

I am not quite sure what conclusion to draw from the above argument. Maybe it offers some support for causal finitism, by suggesting that things are weird when you have a backwards infinite causal sequence?

Causal Robinson Arithmetic

Say that a structure N that has a distinguished element 0, a unary function S, and binary operations + and ⋅ is a causal Robinson Arithmetic (CRA) structure iff:

1. The structure N satisfies the axioms of Robinson Arithmetic, and

2. For any x in N, x is a partial cause of the object Sx.

The Fundamental Metaphysical Axiom of CRA is:

• For every sentence ϕ in the language of arithmetic, ϕ is either true in every metaphysically possible CRA structure or false in every metaphysically possible CRA structure.

Causal Finitism—the doctrine that nothing can have infinitely many things causally prior to it—implies that any CRA is order isomorphic to the standard natural numbers (for any element in the CRA structure other than zero, the sequence of predecessors will be causally prior to it, and so by Causal Finitism must be finite, and hence the number can be mapped to a standard natural number), and hence implies the Fundamental Metaphysical Axiom of CRA.

Given the Fundamental Metaphysical Axiom of CRA, we have a causal-structuralist foundation for arithmetic, and hence for meta-mathematics: We say that a sentence ϕ of arithmetic is true if and only if it is true in all metaphysically possible CRA structures.

More on God causing infinite regresses

In my previous two posts I focused on the difficulty of God creating an infinite causal regress of indeterministic causes as part of an argument from theism to causal finitism. In this post, I want to drop the indeterministic assumption.

Suppose God creates a backwards infinite causal regress of (say) chickens, where each chicken is caused by parent chickens, the parent chickens by grandparent chickens, and so on. Now, I take it that the classical theist tradition is right that no creaturely causation can function without divine cooperation. Thus, every case where a chicken is caused by parent chickens is a case of divine cooperation.

Could God’s creative role here be limited to divine cooperation? This is absurd. For then God would be creating chickens by cooperating with chickens!

So what else is there? One doubtless correct thing to say is this: God also sustains each chicken between its first moment of life and its time of death. But this sustenance doesn’t seem to solve the problem, because the sustenance is not productive of the chickens—it is what keeps each chicken in existence after it has come on the scene. So while there is sustenance, it isn’t enough. God cannot create chickens by cooperating with chickens and by sustaining them.

Thus God needs to have some special creative role in the production of at least some of the chickens, fulfilling a task over and beyond cooperation and sustenance. Furthermore, this special task must be done by God in the case of an infinite number of the chickens, since otherwise there would be a time before which that task was not fulfilled—and yet God created infinitely the chickens before that time, too, since we’re assuming an infinite regress of chickens.

What happens in these cases? One might say is that in these special cases, God doesn’t cooperate with the parent chickens. But since no creaturely causation happens without divine cooperation, in these cases the parent chickens don’t produce their offspring, which contradicts our assumption of the chickens forming a causal regress. So that won’t do.

So in these cases, we seem to have two things happening: divine cooperation with chicken reproduction and divine creation of the chicken. Since divine cooperation with chicken reproduction is sufficient to produce the offspring, and divine creation of the chicken is also sufficient, it follows that in these cases we have causal overdetermination.

Now, we have some problems. First, does this overdetermination happen in all cases of chicken reproduction or only in some? It doesn’t need to happen in all of them, since it is overdetermination after all. But if it happens only in some, then it is puzzling to ask how God chooses which cases he overdetermines and which he does not.

Second, when there is overdetermination, the overdetermination is not needed for the effect. So it seems that if God’s additional role is that of overdetermining the outcome, that role is an unnecessary role, and the chickens could be produced by mere divine cooperation, which we saw is absurd. This isn’t perhaps the strongest of arguments. One might say that while in each particular case the overdetermining divine creative action is not needed, it is needed that it occur in some (indeed, infinitely many) cases.

Third, just as it is obviously absurd if God creates chickens merely by cooperating with chickens, it seems problematic, and perhaps absurd, that God creates chickens merely by cooperating with chickens and overdetermining that cooperation.

Famously, Aquinas thinks that God could have created an infinite regress of fathers and sons, and hence presumably of chickens as well. At this point, I can think of only one plausible way of getting Aquinas out of the above arguments, and it’s not a very attractive way. Instead of saying that God cooperates with the production of offspring, we can say that occasionalism holds in every case of substantial causation, that all causation of one substance’s existence by another is a case of direct divine non-cooperative causation, with the creaturely causation perhaps only limited to the transmission of accidents. Like all occasionalism, an occasionalism about substance causation is unappealing philosophically and theologically.

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.

From theism to causal finitism

Causal Finitism—the thesis that nothing can have an infinite causal history—implies that there is a first cause, and our best hypothesis for what a first cause would be is God. Thus:

1. If Causal Finitism is true, God exists.

But I think one can also argue in the other direction:

2. If God exists, Causal Finitism is true.

Aquinas wouldn’t like this since he thought that God could create a per accidens ordered backwards-infinite causal series.

In this post, I want to sketch an argument for (2). The form of the argument is this.

3. God cannot create a sequence of beings ..., A₋₃, A₋₂, A₋₁, A₀ where each being causes the next one.

4. If God cannot create such a sequence, such a sequence is impossible.

5. The best explanation of the impossibility of such a sequence is Causal Finitism.

Claim (4) comes from omnipotence. Claim (5) is I think the weakest part of the argument. Causal Finitism follows logically from the conjunction of two theses, one ruling out backwards-infinite causal chains and the other ruling out infinite causal cooperation (a precise statement and a proof is given in Chapter 2 of my Infinity book). But I am now coming to think that there is a not crazy view where one accepts the anti-chain part of Causal Finitism but not the anti-cooperation part. However, (a) the main cost of Causal Finitism come from the anti-chain part (the anti-chain part is what forces either a discrete time or a discrete causal reinterpretation of physics), (b) there are significant anti-paradox benefits to maintaining the anti-cooperation part, and (c) the theory may seem more unified in having both parts.

Now let’s move on to (3). Here is an argument. Say that an instance of causation is chancy provided that the outcome has a probability less than one.

6. If God can create a backwards-infinite causal sequence of beings, he can create a backwards-infinite chancy causal sequence of beings as the only thing in creation.

7. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings as the only thing in creation, then there is no creature x such that God determines x to exist.

8. Necessarily, if God creates, he acts in a way that determines that something other than God exists.

9. Necessarily, if God determines that something other than God exists then there is a creature x that God determines x to exist.

10. Necessarily, if God creates a backwards-infinite chancy causal sequence of beings, then there is a creature x such that God determines x to exist. (8,9)

11. Hence, God cannot create a backwards-infinite chancy causal sequence of beings. (7,10)

12. Hence, God cannot create a backwards-infinite causal sequence of beings. (6,11)

The thought behind (6) is an intuition about modal uniformity. I think (6) is probably the most vulnerable part of the argument, but I don’t think it’s the one Aquinas would attack. What I think Aquinas would attack would most likely be (7). I will get to that shortly.

But first a few words about (8). In theory, it is possible to determine that something exists without determining any particular thing to exist. One can imagine a being with a chancy causal power such that if it waves a wand necessarily either a bunny or a pigeon is caused to exist, with the probability of the bunny being 1/2 and the probability of the pigeon being 1/2. But God is not like that. God’s will is essentially efficacious and not chancy. God can play dice with the universe, but only by creating dice. Thus, if God wanted to ensure there is a bunny or a pigeon without ensuring which specific one exists, he would have to create a random system that has chancy propensities for a bunny and for a pigeon and that must exercise one of the two propensities.

In fact, I think divine simplicity may imply this. For by divine simplicity, any two possible worlds that differ must differ in something outside God. Now consider a world w₁ where God determines a bunny to exist, and a world w₂ where God merely determines that a bunny or a pigeon exists and in fact a bunny is what comes about. There seems to be no difference outside God between these two worlds (one might wonder about the relation of being-created: could there be an relation of being-created-chancily and being-created-non-chancily? this seems fishy to me, and suggests a regress—how are the two relations differently related to God? and do we want to multiply such relations, saying there is such a thing as being-created-chancily-with-probability-0.7?). If both worlds are possible, by divine simplicity they must be the same, which is absurd. So at least one must be impsosible. And w₂ is a better candidate for that than w₁.

That still doesn’t establish (8). For I admitted that God can play dice if he creates dice. Thus, it seems that God could determine that something exists without determining where it’s A or B or C (say) by determining there to be dice that decide whether A or B or C are produced. But on this story, God still determines there to be dice, so there is an x—a die—that God determines to exist. I think a bit more could be said here, but as I said, I don’t think this is the main thing Aquinas would object to.

Back to (7). Why can’t God create a chancy backwards-infinite causal sequence while determining some item A_n in it to exist? Well, the sequence is chancy, so the probability that A_n − 1 causes A_n given that A_n − 1 exists is some p < 1. But, necessarily, if one creature causes another, it does so with divine cooperation (Aquinas will not disagree), and conversely if God cooperates with one creature to cause another, the one creature does cause the other. That the probability that God cooperates with A_n − 1 to cause A_n is equal to the probability that A_n − 1 causes A_n, because necessarily one thing happens if and only if the other does. Thus, the probability that God cooperates with A_n − 1 to cause A_n, given that A_n − 1 exists, is p. But p < 1, so it sure doesn’t look like a case of God determining A_n to exist!

But perhaps there is something like overdetermination, but between determination and chanciness (so not exactly over-determination). Perhaps God both determines A_n to exist and chancily cooperates with A_n − 1 to produce A_n. One problem with this hypothesis is with divine simplicity: it does not seem that there is any difference outside God between a world where God does both and God merely cooperates or concurs. But Aquinas may respond: “Yes, exactly. Necessarily, when one creature chancily causes another, God’s primary causation determines which specific outcome results. Thus there is no world where God merely cooperates.” So now the view is that whenever we have chancy causation, necessarily God determines the outcome. But suppose I chancily toss a coin, and it has chance 1/2 of heads and chance 1/2 of tails. Then on this view, I get heads if and only if God determines that I get heads. Hence the chance that God determines I get heads is 1/2. But it seems plausible that God’s determinations are not measured by numerical probabilities, and in any case that they are not measured by numerical probabilities coming from our world’s physics!

Grim Toe-Cutters

Imagine that Fred has all ten toes at 10 am, and there are infinitely many grim reapers. When a grim reaper wakes up, it looks at Fred, and if he has all his ten toes, it cuts one off and destroys it; otherwise, it does nothing. There are no other toe-cutters around.

Suppose, further, that grim reaper wake-up times can be set by you to any times between 10 and noon, endpoints not included. If you set the activation times to be such that there is a first activation time after 10 am (e.g., the nth reaper wakes up 60/n minutes before noon), there is no paradox of any sort. But if you set the times such that they are all after 10 am, but before every activation time there is another activation time, then… well, then logic guarantees that Fred will get a toe cut off infinitely many times and will regrow a toe infinitely many times! For without toe-regrowing, we get a paradox.

This is, of course, logically and metaphysically possible. Toes can regrow, and it is metaphysically though perhaps not physically possible for them to do so quickly. But what is amazing is that just by setting wake-up times for grim toe-cutters, we can make this miracle happen.

Grim Reapers and logical impossibility

The main objection to the Grim Reaper paradox as an argument against infinite causal sequences is the Unsatisfiable Pair (UP) objection that notes that paradox sets up an impossible situation—and that’s why it’s impossible!

I’m exploring a response that distinguishes metaphysical and (narrowly) logical unsatisfiability. The Grim Reaper situation is not logically unsatisfiable. The UP objection (well, really, Unsatisfiable Quadruple) notes that the following cannot all be true:

1. For all n > 0, the nth reaper wakes up at 60/n minutes after 10 am and kills Fred if and only if Fred is alive.

2. Fred is alive at 10 am.

3. There are no possible causes of Fred’s death other than those described in (1).

4. There are no possible causes of Fred’s resurrection.

But all that’s needed to have these four claims hold is for each reaper to kill Fred and then have Fred causelessly come back to life before the next one kills him. And while I think causeless resurrections are metaphysically impossible, they are (narrowly) logically coherent.

In other words, for the UP objection to work, the unsatisfiability must be metaphysical, not merely narrowly logical. But this, I think, negatively affects the force of the UP objection. For instance, in my Infinity book I consider Grim Reapers with adjustable wake-up times, and I note that for some wake-up time settings (say, the nth reaper wakes up 60/n minutes before noon) there is no paradox, and I ask what metaphysical force prevents the wake-up time settings from being the paradoxical ones. Daniel Rubio in a review of the book responds (in the context of a parody) that “no metaphysical thesis is required to explain this impossibility; the fact that it would lead to a contradiction is enough.” But in fact a metaphysical thesis is required to explain the impossibility, since there is no contradiction (in the narrowly logical sense) in (1)–(4).

Perhaps this is not a big deal. After all the metaphysical thesis here, that causeless events are impossible, is one that I do accept. But nonetheless it is a metaphysical thesis, as such on par with causal finitism, and hence when we consider the explanation of the impossibility of the Grim Reaper story and the impossibility of various other of the causal paradoxes that I discuss, there is something appealing about seeing the case as nonetheless offering support for causal finitism, which explains all of them, while the thesis about causeless events being impossible does not.

Unreliable Grim Reapers

As usual, Fred is alive at 10 am, and there is an infinite sequence of Grim Reapers, where the nth has an alarm set for 60/n minutes after 10 am, and if the alarm goes off, it checks if Fred is dead, and swings its scythe at Fred if and only if Fred is alive. But here’s the twist. These Grim Reapers are unreliable killers. The probability that the nth Reaper’s swing would succeed in killing Fred is 1/n^p, where p is some positive real number, the same for each Reaper, and independently of all other relevant events.

Here’s the fun thing. It seems possible for Fred to survive the whole ordeal. All it takes is for every Grim Reaper to fail at killing Fred. Nothing absurd happens then. Moreover, it seems this isn’t the only way for absurdity to be avoided in this case. We could also suppose that the nth Reaper kills Fred, while Reapers n + 1, n + 2, … all fail.

Suppose we adopt what seems the best alternative to Causal Finitism, namely the Inconsistent Pair response to the original Grim Reaper paradox, which says that the reason the original paradox is impossible is simply because it embodies an Inconsistent set of propositions—some Reaper has to kill Fred and none can. If that’s what’s wrong with the original Grim Reaper paradox, then it seems we have to accept my Unreliable Reaper story as possible.

But things are a little bit more complicated. The only way to avoid paradox in the Unreliable Reaper story is if there is some n ≥ 0 such that all the Reapers starting with Reaper n + 1 fail. But now suppose that 0 < p ≤ 1. Then the event that all the Reapers starting with Reaper n + 1 fail is less than or equal to (1−1/(n+1)^p)(1−1/(n+2)^p)(1−1/(n+3)^p)... = 0 (this is because Σ_k 1/k^p = ∞ if p ≤ 1). Thus the probability that we have avoided paradox is 0. Hence, if we have to avoid paradox, a specific zero probability event—namely, the event of paradox-avoidance—has to happen (the probability of a countable disjunction of zero probability events is zero). But if it has to happen, it can’t be probability zero, but must be probability one!

Perhaps here we bring back the Inconsistent Pair response. We say that my Unreliable Reaper story is impossible if p ≤ 1, because if p ≤ 1, then a zero probability event has probability one, which is inconsistent. No such problem occurs if p > 1. Thus, on this version of the Inconsistent Pair response, my Unreliable Reaper story is impossible if the success probability of the nth Reaper is 1/n^p for p ≤ 1 but possible if p > 1. And that’s pretty counterintuitive.

On finitistic addition

By a finite alphabet encoding of a set X, such as the real numbers, I mean a one-to-one function ψ from X to countably infinite sequences s₀s₁... taken from some finite alphabet. For instance, standard decimal encoding, with a decision whether to have infinite sequences of trailing nines or not, is a finite alphabet encoding of the reals, with the alphabet consisting of ten digits, a decimal point and a sign. Write ψ_k(x) for the kth symbol in the encoding ψ(x) of x.

A function f from Rⁿ to R is finitistic with respect to a finite alphabet encoding ψ provided that there is a function h from the natural numbers to the natural numbers such that the value of ψ_k(f(x₁,...,x_n)) depends only on the first h(k) symbols in each of ψ(x₁), ..., ψ(x_n).

This concept is related to concepts in “real computation”, but I am not requiring that the finite dependences be all implemented by the same Turing machine.

Theorem: Let X be any infinite divisible commutative group. Then addition on X is not finitistic with respect to any finite alphabet encoding.

A divisible group X is one where for every x ∈ X there is a y such that ny = x. The real numbers under addition are divisible. So are the rationals. So is the set of all rotations in the plane.

This has a somewhat unhappy consequence for Information Processing Finitism. If reality encodes real numbers in a discrete way consistent with IPF, we should not expect each real number to have a uniquely specified encoding.

Proof of Theorem: Suppose addition is finitistic with respect to ψ. Let F be the algebra on X generated by the sets of the form {x : ψ_k(x) = α}. If addition is finitistic, then for any A ∈ F, there is a finite sequence of pairs (A₁,B₁), ..., (A_N,B_N) of sets in F such that

1. {(x,y) : x + y ∈ A} = ⋃_i(A_i×B_i).

Therefore:

2. x + y ∈ A if and only if x ∈ ⋃{A_i : y ∈ B_i}.

Thus:

3.  − y + A =  ∪ {A_i : y ∈ B_i}.

Now as y varies over the members of X, there are at most 2^N different sets generated by the right hand side. Thus,  − y + A can take on only finitely many values. Hence, A has only finitely many translates.

But this is impossible. Let Z be the set of x such that x + A = A. This is an additive subgroup of X. Note that x + Z = y + Z iff x − y ∈ Z iff (x−y) + A = A iff x + A = y + A. Thus, if there are only finitely many x + A, there are finitely many x + Z. Hence X/Z is a finite group. Let n be its order. Then n[x] = 0 for every coset [x] = x + Z in R/Z. For any x ∈ X choose y such that ny = x. Then n[y] = 0, and so [x] = 0, thus Z = X. It follows that A is invariant under every translation, so it must be either ⌀ or X. Hence |F| ≤ 2, which is absurd since F is infinite as X is infinite and ψ is one-to-one.

(I got the main idea for this proof from the answer here.)

Information Processing Finitism, Part II

In my previous post, I explored information processing finitism (IPF), the idea that nothing can essentially causally depend on an infinite amount of information about contingent things.

Since a real-valued parameter, such as mass or coordinate position, contains an infinite amount of information, a dynamics that fits with IPF needs some non-trivial work. One idea is to encode a real-valued parameter r as a countable sequence of more fundamental discrete parameters r₁, r₂, ... where r_i takes its value in some finite set R_i, and then hope that we can make the dynamics be such that each discrete parameter depends only on a finite number of discrete parameters at earlier times.

In the previous post, I noted that if we encode real numbers as Cauchy sequences of rationals with a certain prescribed convergence rate, then we can do something like this, at least for a toy dynamics involving continuous functions on between 0 and 1 inclusive. However, an unhappy feature of the Cauchy encoding is that it’s not unique: a given real number can have multiple Cauchy encodings. This means that on such an account of physical reality, physical reality has more information in it than is expressed in the real numbers that are observable—for the encodings are themselves a part of reality, and not just the real numbers they encode.

So I’ve been wondering if there is some clever encoding method where each real number, at least between 0 and 1, can be uniquely encoded as a countable sequence of discrete parameters such that for every continuous function f from [0,1] to [0,1], the value of each parameter discrete parameter corresponding to of f(x) depends only on a finite number of discrete parameters corresponding to x.

Sadly, the answer is negative. Here’s why.

Lemma. For any nonempty proper subset A of [0,1], there are uncountably many sets of the form f⁻¹[A] where f is a continuous function from [0,1] to [0,1].

Given the lemma, without loss of generality suppose all the parameters are binary. For the ith parameter, let B_i be the subset of [0,1] where the parameter equals 1. Let F be the algebra of subsets of [0,1] generated by the B_i. This is countable. Any information that can be encoded by a finite number of parameters corresponds to a member of F. Suppose that whether f(x) ∈ A for some A ∈ F depends on a finite number of parameters. Then there is a C ∈ F such that x ∈ C iff f(x) ∈ A. Thus, C = f⁻¹[A]. Thus, F is uncountable by the lemma, a contradiction.

Quick sketch of proof of lemma: The easier case is where either A or its complement is non-dense in [0,1]—then piecewise linear f will do the job. If A and its complement are dense, let (a_n) and (b_n) be a sequence decreasing to 0 such that both a_n and b_n are within 1/2^n + 2 of 1/2ⁿ, but a_n ∈ A and b_n ∉ A. Then for any set U of positive integers, there will be a strictly increasing continuous function f_U such that f_U(a_n) = a_n if n ∈ U and f_U(b_n) = a_n if n ∉ U. Note that f_U⁻¹[A] contains a_n if and only if n ∈ A and contains b_n if and only if n ∉ A. So for different sets U, f_U⁻¹[A] is different, so there are continuum-many sets of the form f_U⁻¹[A].