Is there some sort of a probability problem with a humongous but finite universe?

It’s easy to generate probabilistic paradoxes in a universe (or multiverse) with infinitely many people (e.g., if infinitely many people roll a die, equal numbers of people get 1 as get more than 1, so why think it’s more likely to get more than 1?). But what about a very large but finite universe? I used to think: “The only relevant difference is between finite and infinite. Really big but finite—no problem.” Now I am not so sure.

Paul Heyl measured the gravitational constant G as 6.670 × 10⁻¹¹ m³ kg⁻¹ s⁻², and denote the latter quantity by G₀. Consider two theories:

• H₁: The gravitational constant is between 6.665 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 6.675 × 10⁻¹¹ m³ kg⁻¹ s⁻².

• H₂: The gravitational constant is between 7.676 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 7.686 × 10⁻¹¹ m³ kg⁻¹ s⁻².

It seems obvious that:

1. Heyl’s measurement strongly supports H₁ but does not completely rule out H₂.

But let’s think this through. Suppose Heyl’s evidence is the proposition E which he would express as “I measured G to be G₀.” But, very plausibly, it is an essential property of a human being that they exist in a world with such-and-such a gravitational constant. One way of getting to this conclusion is to say that the forces of gravity are part of our causal history, and then to apply the essentiality of origins. Another is to say that we couldn’t have been made of completely different matter, but the forces exerted by the matter in our bodies are an essential property of that matter.

Given this essentiality of gravitational constant assumption, it follows that at least one of H₁ and H₂ is incompatible with Heyl’s existence. Now, to get (1), we need prior probabilities on which P(H₁|E) > P(H₂|E) > 0. Such prior probabilities will assign a non-zero value to H₁E and to H₂E. But at least one of these two claims is impossible since E entails Heyl’s existence, and a probability assignment that assigns a non-zero value to something impossible is screwed up, and we should be quite suspicious of what we get from it.

We might try to avoid this by using self-locating evidence. But my colleague Yoaav Isaacs has this great paper that gives a pretty strong argument that there isn’t a good way to working with self-locating evidence. So suppose we put this option aside.

Or we might make a distinction between logical impossibility and metaphysical impossibility. I find that suspicious, too.

So, what’s left? Well, here’s one remaining suggestion. Heyl’s evidence is equivalent to the proposition that Heyl measured G to be G₀, a proposition that rigidly refers to Heyl, and hence won’t be compatible with both H₁ and H₂. But we can weaken Heyl’s evidence to something that is compatible with H₁ and H₂, something purely qualitative, like:

• E_Q: A physicist named “Paul Heyl”, who married someone named “Lucy Daugherty”, and who …, measured G to be G₀.

Here, “…” is all the other purely qualitative stuff we know about Paul Heyl, so that E_Q is compatible with both H₁ and H₂.

But now here is a problem. Suppose we live in a vast but finite universe with, say, 10¹⁰¹⁰ people. In such a universe, we might well expect large numbers of people named “Paul Heyl” who satisfy all the conditions in E_Q, including the measurement of G to be G₀, even if in fact G is in the range indicated in H₁ (measurement error!). Thus, P(E_Q|H₂) is close to 1 as is P(E_Q|H₁). Granted, we do have P(E_Q|H₁) > P(E_Q|H₂) > 0. But because the two probabilities are so close to each other, the support E_Q gives to H₁ over H₂ is very slight, and hence we no longer have (1).

It follows that unless we can find some other way of solving the problem that the essentiality of the laws of nature to humans poses for Bayesian reasoning, a fair amount of fundamental physics research would be undercut by a large enough—even if finite—universe.

Of course, maybe we can find some other way of solving it. But maybe we can’t. And if we can’t, then the E_Q solution might be our best bet—and it’ll work just fine in a universe that isn’t too vast.

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.)

A method for living forever

Maybe you have a cancer that would kill you in three months.

So, get a powerful rocket.

Accelerate close to the speed of light, and make a one light-year round-trip journey that from your reference frame takes about a month, but takes slightly over a year from the point of view of the earth. If your speed during the first journey was v₁, now repeat the same trip with a speed of v₂ = (3c²+v₁²)^1/2/2. Then repeat with a speed of v₃ = (3c²+v₂²)^1/2/2. And so on, forever.

Fact: Each journey will take a bit more than a year of earth-time but only half of the you-time of the previous. So the total you-time of your journeying will be 1 + 1/2 + 1/4 + 1/8 + ... = 2 months. You’ll never die. At every future time, you will be alive.

But this is pointless. You might as well stay on earth, and then you’ll have three months of you-time. Three months of you-time followed by death is better than two months of you-time with no death.

A Christian argument against eternalism, with some remarks on "finite" and "infinite"

1. We have an infinite future.

2. If eternalism is true, then anything that has an infinite future is infinite.

3. We are finite.

4. So, eternalism is not true.

The crucial premise is 2. One thought behind 2 is that our best version of eternalism holds that we four-dimensional, and if we have an infinite future, that makes us infinite in the fourth dimension.

But I think we can do better than that. Plausibly, part of what we mean by “We have an infinite future” is that we will have infinitely many token future mental states (if not, add that to the premises). On eternalism, all these mental states exist. And they are clearly all ours. So if we have an infinite future, we have an infinite mental life, and that is a way of being infinite.

I am an eternalist, and I want to affirm 1 and 3. What can I do? One move is this. The relevant sense of “finite” in 3 is not a mathematical sense, but something more “metaphysical” like limited. Now, to be limited is to have one or more limits. This is quite compatible with there being respects in which we lack a limit. Thus, the charged infinite rod that sometimes figures in physics homework has limits: not limits of length, but limits of width and height (and others). In the metaphysical sense, then, the rod is finite. Likewise, then, even if we are temporally infinite or infinite in the number of mental states, we are still limited in other ways.

If we go for this move, we have to make a choice what to mean by “infinite”. We could say that something is infinite provided there is some respect in which it is unlimited. If we did that, then one thing could be finite and infinite—as long as it is limited in one way and unlimited in another. The “infinite rod” would then be both finite and infinite. And, if eternalism is true and there is an eternal afterlife, we are finite and infinite. On this take, the argument is invalid, because it is missing the assumption that nothing is both finite and infinite.

A second otion is to make “infinite” mean unlimited in all respects. In that case, we are finite and not infinite. Indeed, only God is infinite then. A set with what the mathematician calls “infinite cardinality” is limited by not having a greater cardinality than the one it has.

A third option would be to take “finite” to mean limited in every way, “infinite” to mean unlimited in all respects, and then allow for the possibility of things that are neither finite or infinite—perhaps us.

A double lottery and non-normalized probabilities

Suppose a positive integer N is generated by a fair lottery.

Then, a random integer K is chosen between 1 and N (inclusive).

What information does this give you about N?

Obviously you now know that N ≥ K. Anything else?

Consider some specific pair of numbers n ≥ k, and suppose we’ve found out that K = k. What’s the probability that N = n? Of course P(N=n|K=k) = 0/0. But what if we do this as a limiting procedure. Suppose first that N is randomly chosen between 1 and M where M ≥ n, and let P_M be the probabilities for this case. Then

• P_M(N=n|K=k) = (1/M)(1/n)/[(1/M)Σ_j=k^Mj⁻¹] = (1/n)/Σ_j=k^Mj⁻¹.

Take the limit as M goes to infinity. Since Σ_n=k^∞j⁻¹ = ∞, the limit is zero, so we don’t have a meaningful distribution for N.

On the other hand, what if we independently choose two random integers K₁ and K₂ between 1 and N? Suppose n ≥ k_i for i = 1, 2. Let k^* = max (k₁,k₂). Then:

• PM(N=n|K1=k1,K2=k2) = (1/M)(1/n2)/[(1/M)Σj=k*Mj−2] = (1/n2)/Σj=k*Mj−2.

Take the limit as M → ∞ and call that P(N=n|K₁=k₁,K₂=k₂). The limit behaves like ck^*/n², for a constant c > 0, and generates a well-defined probability for N = n.

With zero samples, we don’t have a well-defined probability for N. With one sample, we still don’t. But with two samples (or more), now we do. This is a rummy thing: how is it that sampling turns probabilistic nonsense into sense?

This is making me more friendly to using non-normalized probabilities. After all, the fair lottery for N is easily modeled by the constant probability p₀(n) = 1. With one sample N = k, we have p₁(n) = 1/n for n ≥ k and p₁(n) = 0 for n < k. With two samples k₁, k₂, we have p₂(n) = 1/n² for n ≥ max (k₁,k₂) and p₂(n) otherwise. All this makes perfect sense. And there is a lovely mathematical feature of non-normalized probabilities: conditionalization is conjunction. The conditional probability of an event A on event B is just the probability of A ∩ B.

Non-normalized probabilities aren’t going to solve all problems with infinite fair lotteries. For instance, I toss a fair coin and generate a number N with the following rule. On heads, I choose N with my fair lottery on the positive integers. On tails, I choose N such that the probability of N = n is 2⁻ⁿ (e.g., I toss an independent fair coin and let N be the number of the first toss that gives heads). What’s my non-normalized probability p(x,n), where x is heads or tails and n is a positive integer? We surely want ∑_np(H,n) = ∑_np(T,n): the total probability of the heads options equals the total probability of the tails options. But clearly p(T,n) has to exponentially decrease so ∑_np(T,n) is finite and non-zero. On the other hand, p(H,n) is constant, so ∑_np(H,n) is zero or infinity. So they can’t be equal.

But I wonder if one could say something like this: Non-normalized probabilities make sense in certain cases, and in those cases it’s reasonable to use them?

Two decreases in tension between faith and science

Over the past two hundred years or so, one new tension point arose for the relationship between Christianity and science due to scientific progress—namely, evolution. At the same time, several tension points disappeared due two other instances of scientific progress.

The first instance of this scientific progress was the general abandonment of the Aristotelian eternal world model of the universe with Big Bang cosmology. In the middle ages, Jewish, Islamic and Christian thinkers struggled with the tension between the science/philosophy of the day strongly tending towards a universe that always existed and the theological commitment to a creation a finite amount of time ago. That problem is gone.

The second instance is our scientific understanding of the continuity of organic development from zygote to embryo to infant to adult, which has made quite implausible the old view of discontinuous transition in utero from vegetable to animal to human. This old view was the dominant scientific view of human origins until fairly recently, and it had serious tensions with Christian theology.

The first of these embryological tensions was with Christian moral views about abortion. While traditionally Christians opposed both contraception and abortion, abortion was morally seen as a form of homicide. But on the discontinuous transition view, abortion prior to human ensoulment would only be contraception.

The second embryological tension was a technical problem in Christology. Suppose that in the Incarnation we have the vegetable, mere animal and rational animal sequence. Then Aquinas observes there are two possibilities, neither of which is theologically appealing.

First, it could be that God becomes incarnate as a vegetable or a mere animal. But this seems, as Aquinas says, “unbecoming”. And he seem to be right. The Incarnation reveals to us the person of the Logos, and it would be unbecoming that the Logos become a non-personal being.

Second, it could be that the Incarnation happens only at the beginning of the third stage of development, namely once everything is ready for a rational animal. But then Aquinas says “the whole conception could not be attributed to the Son of God”. Indeed, don’t we even have a tension with the Apostles’ Creed line that Christ “was conceived by the Holy Spirit”? For on this option, Christ was not conceived at all. What was conceived was a vegetable, not Christ. (Indeed, none of us were conceived on this view.) Moreover, one might worry that then there would be a sense in which the flesh of Christ would pre-exist the Incarnation. And that makes it difficult to say that the Word became flesh—for the flesh that Scripture says he “became” would already in a sense have been there, and one can’t become this flesh, since this flesh already has its own identity. (Granted, there may well be some Aristotelian metaphysics one can do to lessen this last worry.)

Aquinas solves the problem by supposing that Christ is conceived fully formed in Mary’s womb, and hence has the rational soul from the first moment of his existence. But this solution is itself problematic. Absent gradual development from a zygote, is this conception at all? If God were to create an adult human either ex nihilo or out of some pre-existing matter, we would not consider that a conception. But neither should we then consider it a conception if God creates a fully-formed fetus, even if he does that out of the pre-existing matter of Mary. So we still have a problem with the Apostles’ creed’s “was conceived by the Holy Spirit”. Moreover, it seems that this deprives Mary of a significant chunk of her motherhood.

But the problem entirely disappears once we think that the human beings begin their existence at conception. Christ is conceived by the Holy Spirit, presumably in that Mary’s ovum is transformed into a zygote by the infinite power of the Holy Spirit, which zygote is the Christ who then grows in utero like we all do.

(Catholics also note that the new scientific understanding of human embryonic development also helps with the doctrine of Mary’s immaculate conception—for only a rational being can be immaculately conceived, since original sin or freedom from it can only apply to a rational being.)

Quantifying saving infinitely many lives

Suppose there is an infinite set of people, all of them worth saving, and you can save some subset of them from drowning. Can you assign a utility U(A) to each subset A of the people that represents the utility of saving the people in A subject to the following pair of reasonable conditions:

1. If A is a proper subset of B, then U(A) < U(B)

2. If A is a subset of the people, and x is one of the people not in A while I is an infinite set of people not in A, then U(A∪{x}) ≤ U(A∪I)?

The first condition says that it’s always better to add extra people to the set of people you save. The second condition says it’s always at least as good to add infinitely many people to the set of people you save as to add just one. (It would make sense to say: it’s always better to add infinitely many, but I don’t need that stronger condition.)

Theorem. For any infinite set of people, there is no real-valued utility function satisfying conditions (1) and (2), but there is a hyperreal-valued one.

It’s obvious we can’t do this with real numbers if we think of the value of saving n lives as proportional to n, since then the value of infinitely many lives will be ∞ which is not a real number. What’s mildly interesting in the result is that there is no way to scale the values of lives saved in some unequal way that preserves (1) and (2).

Proof: The hyperreal case follows from Theorem 2 here, where we let Ω = Ω′ be the set of people, G be the group of permutations of the set of people that shuffle around only finitely many people, and let U be the hyperreal probability (!) generated by the theorem. For this group is clearly locally finite, and any utility satisfying condition (1) and invariant under G will satisfy (2) (apply invariance to a permutation π be that swaps x and a member of I and does nothing else to conclude that U(A∪{x}) = U(A∪{πx}) which must be less than U(A∪I) by (1)).

The real case took me a fair amount of thought. Suppose we have a real U satisfying (1) and (2). Without loss of generality, the set of people is countably infinite, and hence can be represented by rational numbers Q. For a real number x, let D(x) be the Dedekind cut {q ∈ Q : q < x}. Fix a real number x. Choose any rational q bigger than x. Then for any real y > x we will have D(y) ∖ D(x) infinite, and by (1) and (2) we will have:

3. U(D(x)) < U(D(x)∪{q}) ≤ U(D(y)).

Let b = inf_y > xU(D(y)). It follows that U(D(x)) < b ≤ U(D(y)) for all y > x. Let f(x) be the open interval (D(x),b). Then f(x) and f(y) are disjoint and non-empty for x < y. But the collection of disjoint non-empty open intervals of the reals is always countable. (The quick argument is that we can choose a different rational in each such interval.) So f is a one-to-one function on the reals with countable range, a contradiction.

Notes: The positive part of the Theorem uses the Axiom of Choice (I think in the form of the Boolean Prime Ideal Theorem). The negative part doesn’t need the Axiom of Choice if the set of people is countable (the final parenthetical argument about intervals and rationals ostensibly uses Choice but doesn’t need it as the rationals are well-ordered); in general, the argument of the negative part uses the weak version of the Countable Axiom of Choice that says that every infinite set has a countably infinite subset.

Permanence and meaning

Consider this strong meaning-permanence thesis:

1. There being a permanent end to all humanly relevant events would render all of our present activities meaningless.

And this weak one:

2. There being a permanent end to all humanly relevant events would render some of our present activities meaningless.

Here is a quick and easy argument that both are false. Let’s imagine that we believe in a narrative N where there are humanly relevant events that are go on forever and that render some of our present activities meaningful. After all, if there is no such narrative, then it is odd to say that a permanent end to humanly relevant events renders some or all of our present activities meaningless, since these activities would necessarily be meaningless even if there were no such end.

Now, let’s imagine that we came to think that the events and experiences in N exponentially speed up with respect to objective time, in such a way that the first “year”, by human reckoning (revolutions of the earth about the sun, say), described by N takes an objective year, but the second “year” takes half a year, the third “year” takes a quarter of a year, and so on. Thus, we come to think that all the events and experineces in N take place objectively in two years. This is then followed by a clean wipe of reality, and a new creation that has no meaningful connection to any humanly relevant events. Call this story N^*. I think it makes little human difference whether reality is described by N or by N^*. In terms of subjective time, the humanly relevant events of N^* take infinitely long. The only difference is that after the humanly relevant events there are other events that are not humanly relevant. Enriching reality with these events surely does not take away meaning.

So, none of our present activities lose meaning on N^*. But on N^* there is a permanent end of humanly relevant events. Thus, (1) and (2) are both false.

Perhaps this was too quick, though. What if your life project is to fill as much of time with humanity as you can? Then on N, if there are humans always, your project is successful, But on N^*, your project is not successful, because there is infinite humanless time after the end of humanity in two objective years, and so humans occupy only an infinitesimal fraction of time.

But I think it’s mistaken to think that it should be our project to fill up time or space with humans or human events. In other words, the filling-up project is meaningless regardless of success. Take the spatial analogue. Suppose somehow we didn’t know about other galaxies (maybe there are dust clouds shielding them from our view) and we have filled up our galaxy with humans. Would we lose any real meaning in our activities if we found out that reality is richer than we thought, and contains other galaxies beyond our reach? I don’t think so.

The above argument is compatible with a modified version of (1):

3. There being a permanent end to all humanly relevant events after a finite number of events would render all of our present activities meaningless.

For we might think that the reason ordinary stories about a permanent end have a tendency to make us think our activities are meaningless does not have to do with time, but with the idea that the narrative structure for humans requires infinity.

Discrete time and Aristotle's argument for an infinite past

Aristotle had a famous argument that time had no beginning or end. In the case of beginnings, this argument caused immense philosophical suffering in the middle ages, since combined with the idea that time requires change it implies that the universe was eternal, contrary to the Jewish, Muslim and Christian that God created the universe a finite amount of time ago.

The argument is a reductio ad absurdum and can be put for instance like this:

1. Suppose t₀ is the beginning of time.

2. Before t₀ there is no time.

3. It is a contradiction to talk of what happened before the the beginning of time.

4. But if (1) is true, then (2) talks of what is before the beginning of time.

5. Contradiction!

It’s pretty easy to see what’s wrong with the argument. Claim (2) should be charitably read as:

• Not (before t₀ there is time).

Seen that way, (2) doesn’t talk about what happened before t₀, but is just a denial that there was any such thing as time-before-t₀.

It just struck me that a similar argument could be used to establish something that Aristotle himself rejects. Aristotle famously believed that time was discrete. But now argue:

6. Suppose t₀ and t₁ are two successive instants of time.

7. After t₀ and before t₁ there is no time.

8. It is a contradiction of what happened when there is no time.

9. But if (7) is true, then (7) talks of what is when there is no time.

10. Contradiction!

Again, the problem is the same. We should take (7) to deny that there is any such thing as time-after-t₀-and-before-t₁.

So Aristotle needed to choose between his preference for the discreteness of time and his argument for an infinite past.

Entailment and Open Future views

This is probably an old thing that has been discussed to death, but I only now noticed it. Suppose an open future view on which future contingents cannot have truth value. What happens to entailments? We want to say:

1. That Jones will freely mow the lawn tomorrow entails that he will mow the lawn tomorrow

and to deny:

2. That Jones will freely mow the lawn tomorrow entails that he will not mow the lawn tomorrow.

Now, a plausible view of entailment is that:

3. p entails q if and only if it is impossible for p to be true while q is false.

But if future contingents cannot have truth value, then that Jones will freely mow the lawn tomorrow cannot be true, and hence by (3) it entails everything. In particular, both (1) and (2) will be true.

Presumably, the open futurist who believes future contingents cannot have truth value will give a different account of entailment, such as:

4. p entails q if and only if there is no history in which p is true and q is false.

But what is a history? Here is a possible story. For a time t, let a t-possibility be a maximal set of propositions that could all be true together at t. Given the open future view we are exploring, a t-possibility will not include any propositions reporting contingent events after t. If t₁ < t₂, and A₁ is a t₁-possibility while A₂ is a t₂-possibility, we can say that A₁ is included in A₂ provided that for any proposition p in A₁, the proposition that p was true at t₁ is a member of A₂. We can then say that a history h is a function that assigns a t-possibility h(t) to every time t such that h(t₁) is included in h(t₂) whenever t₁ < t₂.

(Technical note: Open theism implies a theory of tensed propositions, I assume. Thus if A is a t₁-possibility, then it is not a t₂-possibility if t₂ ≠ t₁, since any t-possibility will include the proposition that t is present.)

But what does it mean to say that a proposition p is true in a history h. Here is a plausible approach. Suppose t₀ is the present time. Given a proposition p that says that s, let p_t0 be the backdated proposition that at t₀ it was such that s (with whatever shifts of tense are needed in s to make this grammatical). Then p is true in h provided that there is a time t₁ > t₀ such that p_t0 is a member of h(t₁). In other words, a proposition p is true in h provided that eventually h settles its truth value.

This works nicely for letting us affirm (1) and deny (2). In every history in which it becomes true that Jones will freely mow the lawn it becomes true that Jones will mow the lawn, while this is not so if we replace the consequent with “Jones will not mow the lawn.” But what about statements that quantify over times? Consider:

5. Jones will mow the lawn, and for every time t at which Jones will mow the lawn, there will be a time t′ that is more than a year after t such that Jones will freely mow the lawn at t′.

This entails:

6. Jones will mow the lawn, and for every time t at which Jones will mow the lawn, there will be a time t′ that is more than a year after t such that Jones will mow the lawn at t′.

but does not entail:

7. Jones will not mow the lawn.

But there is no history h at which (5) is true by the above account of truth-at-a-history given our open future view. For let t₀ be the present and let p be the proposition expressed by (5). Then at any future time t and any history h, the proposition p_t0 is not a member of h(t). For if it were a member of h(t), it would be affirming the existence of an infinite number of future free mowings, and such a proposition cannot be true on our open future view. Since there is no history h at which (5) is true, by (4) we have it that (5) entails both (6) and (7), which is the wrong result.

What if instead of saying that future contingents lack truth value, we say that they are all false? This requires a slight modification to the account of p being true at a history. Instead of saying that p is true at h provided that there is some future time t such that p_t0 is in h(t), we need to say that there is some future time t such that p_t0 is in h(t′) for all t′ ≥ t. This gives the right truth values for (1) and (2), but it also makes (7) true.

I think the above open futurist accounts of entailment work nicely for statements with a single unbounded quantifier over times, but once we get alternating quantifiers like in (5), where the second conjunct is of the form ∀t∃t′ϕ, things break down.

Perhaps the open futurist just needs to be willing to bite the bullet and say that (5) entails (7)?

Combining epistemic utilities

Suppose that the right way to combine epistemic utilities or scores across individuals is averaging, and I am an epistemic act expected-utility utilitarian—I act for the sake of expected overall epistemic utility. Now suppose I am considering two different hypotheses:

• Many: There are many epistemic agents (e.g., because I live in a multiverse).

• Few: There are few epistemic agents (e.g., because I live in a relatively small universe).

If Many is true, given averaging my credence makes very little difference to overall epistemic utility. On Few, my credence makes much more of a difference to overall epistemic utility. So I should have a high credence for Few. For while a high credence for Few will have an unfortunate impact on overall epistemic utility if Many is true, because the impact of my credence on overall epistemic utility will be small on Many, I can largely ignore the Many hypothesis.

In other words, given epistemic act utilitarianism and averaging as a way of combining epistemic utilities, we get a strong epistemic preference for hypotheses with fewer agents. (One can make this precise with strictly proper scoring rules.) This is weird, and does not match any of the standard methods (self-sampling, self-indication, etc.) for accounting for self-locating evidence.

(I should note that I once thought I had a serious objection to the above argument, but I can't remember what it was.)

Here’s another argument against averaging epistemic utilities. It is a live hypothesis that there are infinitely many people. But on averaging, my epistemic utility makes no difference to overall epistemic utility. So I might as well believe anything on that hypothesis.

One might toy with another option. Instead of averaging epistemic utilities, we could average credences across agents, and then calculate the overall epistemic utility by applying a proper scoring rule to the average credence. This has a different problematic result. Given that there are at least billions of agents, for any of the standard scoring rules, as long as the average credence of agents other than you is neither very near zero nor very near one, your own credence’s contribution to overall score will be approximately linear. But it’s not hard to see that then to maximize expected overall epistemic utility, you will typically make your credence extreme, which isn’t right.

If not averaging, then what? Summing is the main alternative.

Closed time loop

Imagine two scenarios:

1. An infinitely long life of repetition of a session meaningful pleasure followed by a memory wipe.

2. A closed time loop involving one session of the meaningful pleasure followed by a memory wipe.

Scenario (1) involves infinitely many sessions of the meaningful pleasure. This seems better than having only one session as in (2). But subjectively, I have a hard time feeling any preference for (1). In both cases, you have your pleasure, and it’s true that you will have it again.

I suppose this is some evidence that we’re not meant to live in a closed time loop. :-)

Shuffling an infinite deck

Suppose infinitely many blindfolded people, including yourself, are uniformly randomly arranged on positions one meter apart numbered 1, 2, 3, 4, ….

Intuition: The probability that you’re on an even-numbered position is 1/2 and that you’re on a position divisible by four is 1/4.

But then, while asleep, the people are rearranged according to the following rule. The people on each even-numbered position 2n are moved to position 4n. The people on the odd numbered positions are then shifted leftward as needed to fill up the positions not divisible by 4. Thus, we have the following movements:

• 1 → 1

• 2 → 4

• 3 → 2

• 4 → 8

• 5 → 3

• 6 → 12

• 7 → 5

• 8 → 16

• 9 → 6

• and so on.

If the initial intuition was correct, then the probability that now you’re on a position that’s divisible by four is 1/2, since you’re now on a position divisible by four if and only if initially you were on a position divisible by two. Thus it seems that now people are no longer uniformly randomly arranged, since for a uniform arrangement you’d expect your probability of being in a position divisible by four to be 1/4.

This shows an interesting difference between shuffling a finite and an infinite deck of cards. If you shuffle a finite deck of cards that’s already uniformly distributed, it remains uniformly distributed no matter what algorithm you use to shuffle it, as long as you do so in a content-agnostic way (i.e., you don’t look at the faces of the cards). But if you shuffle an infinite deck of distinct cards that’s uniformly distributed in a content-agnostic way, you can destroy the uniform distribution, for instance by doubling the probability that a specific card is in a position divisible by four.

I am inclined to take this as evidence that the whole concept of a “uniformly shuffled” infinite deck of cards is confused.

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.

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].

Information Processing Finitism

When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. I ended up largely abandoning that approach, partly because of my qualms about the concept of information and perhaps partly because of worries about physics that I will discuss below.

But I still think the alternative, which one might call information processing finitism, is something someone should work out in more detail.

• [IPF] Nothing with finite informational content can essentially causally depend on anything with infinite informational content.

Here, informational content is by definition contingent. The “essentially” excludes cases where finite informational content depends on a finite part of something with infinite informational content. How exactly the “essentially” is spelled out is one thing I am not clear on as yet.

The main difficulty with IPF is that our physics seems to violate it. The exact current temperature in Waco depends on the exact temperature, pressure and other facts around the world yesterday. Each of the latter facts involves infinite information—temperature is quantified with a real number, and a real number contains infinite information. Note that here IPF and CF may diverge. An advocate of CF can say that the exact current temperature in Waco depends on a finite number of past events such as “yesterday particle n has parameters P”, even if the parameters P involve real numbers that have infinite energy.

One way to escape this difficulty is to assume that our fundamental physics is actually discrete, and the real numbers in our equations are just an approximation. But I don’t want to stick my neck out so far.

Let’s see if we can make IPF work out with a continuous dynamics. We can suppose that metaphysically speaking, an entity’s having a real-valued parameter is constituted by the entity’s having an infinite sequence of discrete parameters, which parameters are more ontologically fundamental than the real-valued parameter.

For instance, by a one-to-one mapping we can assume our real number is strictly between zero and one, and then define it as an infinite decimal sequence 0.b₁b₂..., specified by an infinite sequence of digits. Unfortunately, then, we have some severe restrictions on what kind of dynamics we can have if we require that each digit of the output depend only on a finite number of digits of the input. For instance, multiplication by 3/4 cannot be defined, because to know whether f(x) starts with 0.24 or 0.25, you’d have to know whether x < 1/3 or x ≥ 1/3, and if the input is 0.333..., then you can’t tell from a finite number of digits which is the case. This kind of problem will occur with any other base.

It would be really nice to find some way of encoding a real number as an infinite sequence of discrete parameters each of which takes on a fixed finite range that escapes this kind of a problem. I am pretty sure this is impossible, but am too tired to prove it right now.

But there is another approach. We can have non-unique (many-to-one) encodings of reals. Here is one such approach, probably not the most natural one. Consider sequences of natural numbers n₁, n₂, ... such that for all k we have n_k ≤ 2k and there exists a real number x between 0 and 1 inclusive with the property that |x−n_k/2k| ≤ 1/k. Say that such a sequence encodes the real number x. In general, there will be more than one sequence encoding x by this rule.

Then if f is a function from [0,1] to [0,1], if we have a sequence n₁, n₂, ... encoding the real number x, to generate an acceptable kth term in a sequence encoding f(x), it suffices to know f(x) to within precision 1/2k, and if f is continuous, then we can do that by knowing a finite number of terms in a sequence encoding x (this is because every continuos function on [0,1] is uniformly continuous).

So any continuous dynamics from [0,1] to [0,1] can be handled in this way. The cost is that fundamental reality has degrees of freedom that are unimportant physically—for fundamental reality distinguishes between different sequences encoding the same real x, but the difference has no physical significance.

I don’t know if there is a way to do this with a unique encoding.