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.

Determining camera pose from three landmarks and accelerometer: initial notes

Suppose we have three landmarks at known locations in three dimensional space, not all on one line, and a camera that is not located at any landmark but sees all three landmarks, with known optical parameters such as focal length. The problem of determining the camera pose—the camera position and angles—from the positions of the landmarks in the camera image is known as the P3P problem. It is known that in general there will be at most four solutions for camera pose. And indeed sometimes there will be four solutions.

What if we have some additional information, namely we know how the camera is oriented with respect to gravity (e.g., because the camera is held horizontally or it’s equipped with an accelerometer)? Call the problem of reconstructing the camera image from n landmarks and gravity data PnPA. I recently showed that with just two landmarks, i.e., P2PA, there will be either one, two or infinitely many solutions, and geometrically characterized exactly which case occurs when.

Question: What can we say about the number of solutions to P3PA?

In this post I will make some slight progress on this question.

First we characterize when there are more than two solutions. Note that once we know the camera position, we can calculate the direction it’s pointing from the camera image (Lemma 4 in my paper). So we only need to look at the number of solutions for camera position.

Fact 1: There are at most two solutions for P3PA, except in the case where the three landmarks and camera all lie on one horizontal circle, in which case there are infinitely many solutions.

Proof: For P2PA, we have more than two solutions in precisely the following cases: (a) the two landmarks are on a single vertical line; (b) the two landmarks are in the same horizontal plane and so is the camera; and (c) the two landmarks and the camera are all on one line. To have more that two solutions for P3PA, each pair of landmarks must satisfy at least one of (a)–(c). Suppose this is so.

Suppose first that two landmarks, say m₁ and m₂, satisfy (a). Next suppose that no two landmarks lie on the same horizontal plane, so (b) is satisfied for no pair of landmarks. Then the third landmark m₃ does not lie on the same line as both m₁ and m₂, and hence neither the pair m₁ and m₃ nor th epair m₂ and m₃ satisfies (a), and at least one of these pairs fails to satisfy (c). Hence we have a pair that fails to satisfy any of (a)–(c), and we have at most two solutions by my P2PA result.

Now, continuing to suppose m₁ and m₂ satisfy (a), suppose that some pair of landmarks lies on the same horizontal plane. It can’t be m₁ and m₂ (as then they will be at the same point, and hence all three landmarks will be on one line). Without loss of generality, suppose m₁ and m₃ lie on the same horizontal plane H. The pair m₁ and m₃ cannot satisfy (a) (or else m₃ is at the same point as m₁). If it satisfies either (b) or (c), the camera is on the plane P, and hence in any case we have (b).

Furthermore, if no two landmarks satisfy (a), then since it can’t be that every pair of landmarks satisfies (c) as that would put all the landmarks on one line, at least one pair of landmarks must satisfy (b).

We thus have reduced to the case where a pair of landmarks satisfies (b): they are on the same horizontal plane H as the camera C. Let’s say that these landmarks are m₁ and m₂. Let m′₃ be the projection of m₃ to this plane. From the camera’s optical parameters and the camera image, we can calculate the angles m₁Cm₂, m₁Cm′₃ and m₂Cm′₃. It is known that the locus of points in a plane that subtend the same angle to two fixed points is an arc through these points. Thus, if we are to have more than two solutions, the arcs respectively through m₁Cm₂, m₁Cm′₃ and m₂Cm′₃ must intersect in at least three points. This would require m₁, m₂, m′₃ and C to all lie on the same circle T.

Now, suppose m₃ lies off the plane H. Then given the camera image and the gravity vector, we can measure the angle between m₃, the camera and the plane H, and given the position of m₃ we can compute the distance from m′₃ to the camera. This constrains the camera to lie on the circle T as well as on a second circle T′ around m′₃. Since m′₃ lies on T, these two circles intersect in at most two points. Thus, we have at most two solutions.

On the other hand, when m₃ is in the same plane, so m₁, m₂, m₃ and the camera lie on the same horizontal circle, we will have infinitely many solutions. For if A and B are two fixed points on a circle, and C is a third point on the same side as A and B, the angle ACB will be constant regardless of the choice of C. The landmarks m₁, m₂ and m₃ split T into three arcs, and the camera could be anywhere in the arc it’s in as far as the image goes.

Fact 2: There are cases where there are exactly two solutions.

Proof by picture: Suppose m₁ and m₂ lie on the same plane and m₃ lies off the plane. Suppose the camera is horizontally oriented, and pointed at m₁. Wherever the camera is on the blue arc, it sees m₁ and m₂ the same way (because the angle indicated by the dotted lines does not change, as discussed above). The image of m₃ in the camera is always directly above or below the image of m₁, and as long as the distance from camera to m₁ is the same, the image of m₃ does not move. Thus, the image of m₃ does not change as the camera moves on the red circle. Hence, at the two points where the red and blue circles intersect, the camera sees the same thing, and hence we have two solutions.

Against Arianism

1. The Christian’s deepest desire is for direct union with God.

2. Our union with the Father must always be mediated by the Son.

3. So, if only the Father is God, then the Christian’s deepest desire cannot be fulfilled.

4. The Christian’s deepest desire can be fulfilled.

5. So, it is false that only the Father is God.

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.

A tweak to the Aristotelian account of individuation

The standard Aristotelian story about individuation of finite substances is two-fold:

1. What makes x and y distinct when x and y are members of different species is their difference in form

2. What makes x and y distinct when x and y are members of the same species is their difference in matter.

But I think a slight tweak is in order. Suppose Alice and Spot are both members of different species and have a difference of matter. By the standard story, what accounts for their distinction is just their having different form. But it seems right to say that their difference in matter is also sufficient to distinguish them. The distinction between Alice and Spot is overdetermined by a difference in species and a difference in matter.

If so, then intead of (1) and (2), we should have an inclusively disjunctive account:

3. What makes x and y distinct is that they have a difference in form or have a difference in matter or both.

To make this account work even better, I will stipulate that the difference between an immaterial substance (say, an angel) and a material substance (say, a human) also counts as a difference in matter. Thus, there are two sorts of differences of matter: two things having different matter, or one thing having matter and the other not.

More on infinite chances at salvation

Yesterday, I offered a Toy Model of how God could give infinite chances to someone and they could still reject him. The crucial assumption of the model was that it might take multiple good choices to dig oneself out of the moral hole that one’s wickedness got one into.

But one can also have a model without that assumption. Suppose that every day I make a choice for or against God. If I choose for God, I am saved and God makes me virtuous. If I choose against God, I become habituated into a bit more vice. How much more? Well, we can measure a person’s virtue by how likely they are to choose for God. So let’s suppose that there is some number β strictly between 0 and 1 such that each time I choose against God, my goodness—as measured by the probability of my next choice being for God—is multiplied by β. We can think of β as a kind of wicked habituation constant.

What is the probability that I will always reject God? Suppose my initial level of goodness is α. Then the probability that I will reject God on day 1 is 1 − α. If I do that, then on day 2 my level of goodness will be αβ, and the probability that I will reject God will be 1 − αβ. And if I do that, then on day 3 my level of goodness will be αβ², and the probability that I will reject God will be 1 − αβ². So the probability that I will always reject God is:

• P(R_∞) = (1−α)(1−βα)(1−β²α)....

Easy mathematical fact: as long as α < 1 (i.e., I am not maximally good on day 1) and β < 1 (the bad choice does decrease my goodness), P(R_∞) is bigger than zero. In other words, on this model, I have infinite chances, and each choice has the possibility of resulting in my complete redemption, and yet I have a non-zero probability of choosing badly for eternity.

Let’s make up some conservative numbers. Suppose the infinite chances regimen starts with death, and that β = 0.99: one gets 1% worse each time one chooses against God—and that’s 1% of my previous level of goodness, rather than an absolute 1%. Presumably people die with various levels of goodness between 0 and 1. Suppose one in a thousand people dies with a goodness level α ≤ 0.1. Now if one has goodness level α = 0.1 when one starts the infinite chances regimen at death, and β = 0.99, it turns out (or so Mathematica calculates QPochhammer[.1,.99]) that the probability of damnation is about 0.000035. So with 8 billion people, there would be 8 million dying with goodness level at most 0.1, and so we might well expect at least 8, 000, 000 × 0.000035 = 280 of them to be damned even if they got the infinite chances regimen. So on this toy model, we would expect some people now alive to be damned.

Now it seems plausible that it is compatible with God’s goodness that he would allow one in a thousand people to die with a goodness level of 0.1, and that he would allow a character deterioration of at least 1% per posthumous choice against God—this would give people a real ability to affect their character, while yet being very generous with redemptiveness. In other words, even on an infinite chances model, it is compatible with God’s goodness that hundreds of people be forever separated from God. But if hundreds of people damned is compatible with God’s goodness on an infinite chances model, why wouldn’t hundreds of people damned also be compatible with God’s goodness apart from an infinite chances model, as long as it was also a model where God is very generous in redemptiveness and gives people a real ability to affect their character?

A variant of Molinism

The biggest problem with Molinism is the grounding problem: there is nothing in reality wherein the subjunctive conditionals of free will (“middle truths”) could be grounded. As Reuben perceptively noted in a recent comment, the problem is especially big if God is simple, as then the middle truths can’t even be grounded in God.

Anyway, it seems to me that there might be some benefit to Molinists to think about what one might call Individually Grounded Weak Molinism (IGWM). On IGWM, there are no middle truths about the free actions of non-actual persons, but there are middle truths about the free actions of actual persons, which are grounded in properties of these very persons. Thus, “when” (in the explanatory and not temporal sense) God is deciding whether to create Curley, there is no fact of the matter whether Curley would accept a $5000 bribe in circumstances C. But “as soon as” Curley is actual (on eternalism, this will be the case even before Curley temporally exists—maybe as soon as God decides to create Curley), there will be a fact of the matter whether he would accept that bribe in C.

On IGWM, there is an “ontological home” for the middle truths. Moreover, IGWM ameliorates another problem for standard Molinism: the problem of evil. On standard Molinism, God presumably has a vast amount of knowledge that he can use in optimizing the world. E.g., God knows that Adam and Eve would sin, but there is a vast number of couples almost exactly like Adam and Eve, and it seems very unlikely that they would all have sinned in Adam and Eve’s circumstances, so why didn’t God create them? But on IGWM, God only knows that Adam and Eve would sin in the Garden of Even once he decides to create them. One might objet that are presumably a vast number of gardens almost like Eden that God could put Adam and Eve in, and on IGWM God middle-knows how they would behave in each garden, and it would seem surprising if Adam and Eve would have sinned in all of them. Not so. For it might be that there are close correlations between middle-truths about how Eve (say) would act in circumstance C₁ and how she would act in C₂ if C₁ and C₂ are very similar.

But IGWM still has enough bite to yield some of the advantages of Molinism. First, we have the biblical cases. Granted, these could be interpreted in a non-Molinist way, but a Molinist way seems particularly neat. For instance, in the parable of Lazarus and the rich man, God tells the rich man that his brothers wouldn’t repent even if someone came back from the dead to talk to them. On IGWM, this conditional makes sense, and is grounded in the rich man’s brothers. (Granted, this is a parable, so it’s not a prooftext for Molinism.) Second, we still have the nice story about how God can work in our lives and achieve his gracious goals without undercutting our freedom: God knows what we would choose in which circumstances, and can control the circumstances we are in. Third, the Molinist account of efficacious grace works on IGWM with no changes. Fourth, and this is the immediate motivation for my post, my Molinist defense of hell works just fine on this story.

IGWM does not solve all the problems with Molinism. The Adams circularity argument continues to be problematic. Suppose I freely choose A in circumstances C, and God put me in C because he middle knew that I would freely choose A in C. It’s surely note a coincidence that both (a) I freely choose A in C and (b) I would freely choose A in C. There needs to be some kind of explanatory connection between (a) and (b). If (b) is explanatorily prior to (a), then it seems that the middle facts about me explain my actions, and that seems to undercut my freedom. But if (a) is explanatorily prior to (b), then we have a circularity. For (b) is prior to C, and C is prior to (a).

And there may be a way in which the freedom worry about Molinism feels bigger on IGWM. If it’s a property of me that I would freely choose A in C, rather than just being an ungrounded truth, then it looks a bit more like that property constrains me and takes away my freedom.

Infinite chances and a Molinist defense of damnation

Let’s start with a line of thought that doesn’t require Molinism. Let Infinite Chances be the thesis that God never definitively damns someone: even if on some day you have chosen against God, you will still have choices on future days that can get you back to the path of virtue and hence to God.

One might think that Infinite Chances would all but guarantee that everyone is saved in the end. There is no such guarantee, however. Consider this Toy Model. On each future day n, I have level V_n of virtue, negative if I am on balance vicious and positive if I am on balance virtuous. On each day, I make a free choice. The choice will either be good or bad, and will respectively increase or decrease your virtue by one unit. The more virtuous I am, the more likely the choice will be for good, and the more vicious I am, the more likely that it will be for bad, but no matter how vicious I am, there is a non-zero probability that I will choose for good. I also assume that if I am on-balance vicious, the probability that I will choose for good is less than 1/2. Suppose for simplicity that there is some level S > 0 of virtue at which I become definitively saved.

This story is consistent with Infinite Chances. For no matter how vicious I am, there is also a non-zero probability that I will choose the good, thereby decreasing my level of vice by one unit, and the next day choose the good again, and so on, until I reach level S. So I will always have a chance to dig myself out of my vice. But at the same time, it is a mathematical fact that under the above circumstances if I am in fact vicious on day n (i.e., V_n < 0), then there is a non-zero probability that even over an infinite future I will never be better than I am on day n. My next day’s choice might be bad, and then I might bounce up and down, but never dig myself out. Remember that once my level of vice is  − m units, it would take m steps to get back to neutral.

Next, note that the Toy Model seems consistent with God’s goodness. It allows our actions to have genuine effects on our character, and it allows us genuine freedom to mold that character. Yet on this toy model there is a non-zero probability of my being damned. Thus we get a first conclusion:

1. It is consistent with God’s goodness that not everyone is saved.

This isn’t a defense of hell or damnation as it stands, though it is a defense of non-univeralism. I take it (perhaps pace C. S. Lewis) that it is a part of the doctrine of damnation that once one is damned, one doesn’t have a chance to get out of hell. In other words, the doctrine of hell implies a denial of Infinite Chances, why my toy model implies Infinite Chances.

However, given Molinism (I wish Molinism were true, but alas I think it’s not) one can turn this into a defense of damnation. Let’s imagine that God has created a bunch of people in a situation where the Toy Model is true, while with his middle knowledge God knew that given the infinite chances that the Toy Model posits, all but one—let’s call him “Judas”—would eventually be saved. This is compatible with God’s goodness by something like the line of thought that led to (1).

But now imagine that God decided to create this bunch of people, while knowing with his middle knowledge that given the infinite chances of the Toy Model they would be saved except for Judas, but hadn’t yet decided whether to give the people the infinite chances of the Toy Model. Among the things that God middle-knew is that, if he made the Toy Model true, there would a day n on which Judas will have a level of vice which he would never do better than—the next day he would choose badly, and while he might bounce back up a little, he would never be better than V_n. And so God decided that he would freeze Judas morally at the level V_n achieved on day n, taking away Judas’ freedom of choice on subsequent days. As a result, Judas was irredeemably damned on day n, as the doctrine of hell holds.

Would this be compatible with God’s goodness? Yes! Here is why. We have granted that giving Judas the infinite chances of the Toy Model would be compatible with God’s goodness, even if Judas is never saved. But freezing Judas’ moral level seems a better thing to do. For God knows that if Judas’ moral level is frozen at V_n, then Judas won’t get worse, while if Judas had the infinite choices, then he would still never be better than level V_n, and he would be worse than that on at least day n + 1 (since we have supposed that after day n, Judas is never better than V_n, so V_n + 1 is not greater than V_n, and hence it must be less, as it was assumed that all choices either improve or make worse). Since one’s level of virtue or vice is a primary constituent of human wellbeing, this freezing seems a good thing.

I wonder if there is any way to rescue this argument in the absence of Molinism? Maybe. For there may come a time when the probability of Judas ever digging himself out of his vice has gotten so small, while the probability of his digging himself deeper is pretty high, and so it even absent Molinism it might be a mercy not to give Judas more chances.

Libertarian universalism and open futurism

Consider an interesting pair of thesis. First, consider libertarian universalism, which holds that everyone eventually freely—and with the ability to do otherwise—definitively chooses for God. On the grounds of Christian revelation, I think libertarian universalism is false, but I also think it’s clearly logically possible, so it’s only contingently false.

Second, consider open future views of time on which there either are no facts about future contingents or the facts are all-false. Both open future views are logically compatible with the truth of libertarian universalism, but only in a funny sort of way. For it could be that “everyone” (in the sense of the libertarian universalist doctrine, which includes both past, present and future people) has already freely definitively chosen for God. This would require that the end of the world is predestined to come soon, because otherwise the “everyone” includes—or at least possibly includes!—people who have yet to be conceived, and they clearly have yet to definitively choose for God. Moreover, it would require that all the people now alive have already definitively chosen for God, which seems highly implausible.

Let’s suppose that, as seems plausible, that we all agree that some people have yet to already definitively choose for God. Then open futurists cannot affirm libertarian universalism. Can they deny it? Yes, in at least two ways. They might say that God predestined some to non-freely choose for God. Or they might hold that some who have died have already definitively rejected God and that God has predestined them to have no further chances after that rejection.

So open futurists can deny libertarian universalism but cannot affirm it. Can they be open to it? Well, it initially seems hard to say what it means that one is open to a view that one believes is not true. But open futurists can say that although they believe libertarian universalism is not true, it can become true. And that’s a way of being open to it. But a kind of weakish way.

I don’t accept either open future or libertarian universalism, of course.

Hedonic adaptation

Hedonic adaptation is the theory that individual humans have different basic levels of subjective life satisfaction (SLS). When major changes in life circumstances make life change for the better (e.g., marriage) or worse (e.g., divorce or unemployment), life satisfaction goes up and down, but then over five years it creeps back to roughly where it was even if the circumstances do not change back.

Suppose this theory is true. Can we draw any philosophical conclusions from it?

Here is one thought. Suppose the right theory of human well-being is hedonism. It seems plausible to think that SLS measures one’s overall hedonic level averaged over a medium term. If so, then hedonic adaptation makes means we have less reason than we might have thought to seek out gains and to avoid losses in objective life circumstances for ourselves and others. Not that we have none. Short-term pains and pleasures are still real, and major changes in life circumstances do have a genuine effect on the life-time total hedonic level even if after five years much of the change in SLS disappears—and in the case of negative changes, even if much of the change disappears, it probably doesn’t disappear entirely.

Still, it seems plausible that if hedonism is the right theory of human well-being, then the discovery of hedonic adaptation (assuming it is real) should make us, both individually and societally, more passive than is plausible. One’s hedonic level, according to hedonic adaptation, is just not very responsive in the long-term to costly interventions. E.g., is it really worth a society’s putting a lot of effort into preventing unemployment if the unemployed would anyway return to a basic hedonic level, and hedonic level is what matters?

This suggests that hedonic adaptation provides an argument against hedonic theories of well-being, by yielding an implausible consequence of those theories. Now while this argument initially seemed to depend on hedonic adaptation being empirically true, perhaps it does not even require that. All we need, perhaps, is that hedonic adaptation is possibly true, and then we can argue:

1. If well-being is hedonic in nature, then if hedonic adaptation is true, we would have much less reason than we thought for major societal improvements.

But I think we may also find ourselves with the intuition:

2. Even if hedonic adaptation is true, we would not have much less reason than we thought for major societal improvements.

And it arguably follows from 1 and 2, together with

3. It is possible that hedonic adaptation is true

that:

4. Well-being is not hedonic in nature.

(The move from 1–3 to 4 uses modus tollens and two principles in subjunctive conditional logic, namely (a) if p → q and p → r then p → (q&r), and (b) if p is possible and p → q, then q is possible. I think (a) is true, but I am dubious of (b) in general, even though it is true on Lewis’s theory of counterfactuals. However, even if (b) is not true in general, I think it is true in typical cases, and seems likely to be true in the context of my argument agaisnt hedonism.)

In other words, thinking about hedonic adaptation might well shift us to thinking that a major portion of our well-being is more objective than hedonism would make it be. The reason for fighting unemployment isn’t primarily its deleterious effects on subjective life satisfaction, but its (typical) deleterious effects on objective human flourishing.

But there is also another even more speculative consequence of hedonic adaptation. Even if we reject hedonic level or SLS as the measure of human flourishing, it is clear that that we all want SLS, and indeed want long-term SLS. Hedonic adaptation tells us that improving our life-circumstances won’t get us long-term SLS: after half a decade, we’ll go back to where we were. Plausibly, our desire for long-term SLS is a normal and central human desire. Plausibly, normal and central human desires are fulfillable. Yet this desire for long-term SLS is not fulfillable in this life. Hence there is life after death.

In other words, the phenomenon of hedonic adaptation would empirically bolster the argument from desire considered as an argument for an afterlife.

Some Fitch-style arguments

Let strong physicalist supervenience (SPS) be the view that necessarily all truths supervene on first-order truth about the spatiotemporal arrangement of physical stuff.

Here is a plausible thesis:

1. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly known.

Add this premise:

2. If a conjunction is known, so are its conjuncts.

It follows from SPS, 1 and 2 that:

3. Every actual first-order truth about the spatiotemporal arrangement of physical stuff is actually known.

For suppose that p is any actual first-order truth about the spatiotemporal arrangement of physical stuff. Let q be the conjunction of all other actual first-order truths about the spatiotemporal arrangement of physical stuff. Let r be the conjunction of p and q. Then r is a first-order truth about the spatiotemporal arrangement of physical stuff. By 1, there is a possible world w where r is known. Since r contains all actual first-order truths about the spatiotemporal arrangement of physical stuff, w and the actual world have the same first-order truths about the spatiotemporal arrangement of physical stuff. By SPS, it follows that w and the actual world have the same truths. Thus, since it’s true at w that r is known, it’s actually true that r is known. And since r is actually known, so is p by 2. Hence, we have 3.

In exactly the same way, given:

4. Every first-order truth about the spatiotemporal arrangement of physical stuff is possibly explained

5. If a conjunction is explained, so are its conjuncts

and SPS, we get:

6. Every actual first-order truth about the spatiotemporal arrangement of physical stuff actually has an explanation.

But given physicalism, it is clear that 3 and 6 are false. Regarding 3, it is clear that if physicalism is true, nobody knowns how many hairs I have on my head. Regarding 6, given physicalism, there is no explanation of the sum total of physical reality. Thus:

7. Given either 1 and 2, or 4 and 5, at least one of physicalism and SPS is false.

It is tempting to turn this into an argument against physicalism by holding that:

8. If SPS is true, physicalism is true.

But actually I can imagine someone denying 8. Imagine a classical theist who thinks that everything other than God has to be physical. Such a classical theist will actually believe SPS. For on classical theism, facts about God supervene on facts about everything other than God, since classical theism includes divine simplicity according to which there are no contingent facts intrinsically about God, so any two worlds that differ in their facts about God will have to differ in something other than God. And according to our classical theist, everything other than God has to be physical. So any two worlds that differ must differ in facts about the arrangement of physical stuff, which is what SPS says. However, the classical theist is not a physicalist, because God is not physical on classical theism.

Still, I think an argument against the conjunction of physicalism and SPS is still interesting.

My first (and last?) computer science paper

Some years back I was searching for a better method to play lightgun games like Duck Hunt under emulation by using a Wii remote as a light gun.

The standard Wii remote way of pointing at a TV with a sensor bar underneath is not precise enough. The Wii remote has an infrared camera that tracks the LEDs in the sensor bar, and uses that to estimate where the remote is pointing. But since all the LEDs in the sensor bars lie in a single line, there is no way to determine from the camera image exactly how the Wii remote points. (It’s easy to visualize how rotating the Wii remote about the axis defined by the sensor bar won’t change the camera image in it.) The Wii just guesses, but that’s not good enough for lightgun games.

I already had a way that involved putting four infrared LEDs around my TV, but I was wondering if one could do this with a standard sensor bar by leveraging the additional information from the accelerometer in the Wii remote. A web search came up with a paper by d’Alfonso et al. on determining camera pose using two landmarks and accelerometer. The paper claimed a theorem that, except in certain singular cases, one could determine the pose up to a two-option ambiguity—i.e., there will be two camera orientation/position setups that would result in the same landmark images and accelerometer data.

After thinking about the paper, I realized that the claimed theorem was incorrect: there were singular cases the authors didn’t know about, but more interestingly I discovered cases where one could determine camera pose from two landmark images and accelerometer data without any ambiguity. Moreover, one of these cases was precisely the one relevant to me: the two landmarks are on the same horizontal line (the Wii sensor bar supplies precisely that) and the camera is above or below the plane of that line (easy to arrange). Sadly, experiments showed that Wii remote accelerometer precision was insufficient to determine camera pose in this way.

Anyway, I don’t like the idea of there being a published “theorem” that is incorrect. At some point I worked out all the details with trigonometry, and found a precise characterization of the exact conditions under which in theory (i.e., given a sufficiently precise accelerometer, unlike the one in the Wii remote) one could determine camera pose from two landmark images and gravity data with no ambiguity, with an ambiguity of two, or with an ambiguity of infinity (in some singular cases).

I just got a paper with all the details published.

Sadly, as noted, the Wii remote application didn’t pan out. However, I did write some proof of concept code for my Pixel 7 Pro, and found that it worked pretty decently with its accelerometer. Here’s a screenshot where the accelerometer data plus the screen positions of the (centers of) the bottom two Aruco markers, together with data about the camera and the dimensions of my display board, is used to calculate the screen positions of the upper two Aruco markers (the calculated positions are marked in red).

I kind of suspect that the accelerometer in the right-hand Nintendo Switch joycon might be sufficiently precise for use as a lightgun with two infrared LEDs for positioning, but I haven’t had time to implement this. (Part of the problem is that the infrared LEDs that I have at home are the wrong wavelength for the joycon.)

While this may be my last computer science paper, if some computer science grad student or the like is interested in working with me, we could do some empirical work on seeing how applicable the algorithm is.

From one divine person to two or more

1. Personhood is teleologically relational in the sense that a person is the kind of being that necessarily can only be fulfilled in relationships with other persons.

2. Necessarily, any person other than God is a creature.

3. So, if God is a single person, this person necessarily can only be fulfilled in relationships with creatures.

4. If a being only finds its fulfillment in relationships with Fs, then the being’s fulfillment depends on Fs.

5. So, if God is a single person, God’s fulfillment depends on creatures.

6. God’s fulfillment does not depend on creatures.

7. Thus, God is not a single person.

8. thus, if God is a person, God is two or more persons.

(I think most non-Christian monotheists these days will say that God is a single person. But I wonder if it wouldn’t be a better view for them to deny that God is a person at all, and to insist that the centrality of personhood to the understanding of God is a mistaken Christian approach.)

Here is another variant on the argument:

9. Necessarily, a person can only flourish in relationship to another equal person.

10. It is possible that God flourishes.

11. If God is a single person, then it is impossible for there to be another person equal to God.

12. So, God is a single person, it is impossible that God flourishes. (9 and 11)

13. So, God is not a single person. (10 and 12)

14. So, if God is a person, God is two or more persons.

Now, granted, it is a standard view in Christianity that one cannot prove the doctrine of the Trinity by reason. If the arguments above are sound, have I contradicted that view? No, for several reasons. First, the arguments don’t establish that God is three persons. Second, the arguments only establish that if God is a person at all, then God is two or more persons—but a further argument is needed that God is a person at all. (Perhaps perfect being theology can yield that.)

Value supervenes on being

Consider this thesis, which both Augustine and Aquinas would accept:

1. Value supervenes on being (VSB): How valuable reality is is determined by what entities exist.

However:

2. It is good to have a true belief and bad to have a false belief.

We now have an argument against presentism (only present things exist) and growing block (only present and past things exist). Among our beliefs, we have beliefs about contingent future events. Some of these beliefs are contingently true and others are contingently false. (I am assuming a bivalent temporal logic.) The value of these beliefs is determined by what entities exist. But present and past entities are insufficient to determine whether the beliefs are true, and hence what the value of the beliefs is. Thus, contrary to presentism and growing block, there must be future things.