Fictional characters and the ontological argument

This argument is valid:

1. If God exists in any way, he exists in the supreme way of existing.
2. If God does not exist in the supreme way of existing, God exists as a fictional character.
3. So, God exists in the supreme way of existing.

Star Trek cookie cutters for First Contact Day

In honor of First Contact Day, my big kids wanted to make Starfleet insigna and Vulcan IDIC cookies. So I generated some 3D printable cookie cutters (Starfleet; Vulcan).

Yet another odd double effect case

Alice has just fed a poison to Bob. Bob hasn’t died yet. He is standing, by coincidence, on the edge of a cliff, and soon will die of the poison, unless he gets an antidote. Carl is there and has a syringe full of the antidote. Carl injects Bob with the antidote, but this startles Bob and Bob falls off the cliff to his death.

Question 1: Did Alice murder Bob?

Answer: I think not. Here’s an argument. Bob dies as a side-effect of injection with the antidote. But it could just as well have been Carl who slipped and fell while injecting Bob instead of Bob falling. And surely then we shouldn’t say that Alice murdered Carl—though she did wrongfully cause his death.

Question 2: Suppose that Carl was Alice’s friend and foresaw that Bob would fall off the cliff to his death if injected with the antidote, but reasoned: “I am saving Alice from being a murderer.” Could one legitimately make this double-effect analysis? “Carl is intending that Alice not be a murderer. His means to that is giving Bob an antidote to the poison. A foreseen side-effect of Carl’s action is Bob’s death, but this side-effect is not intended either as an end or as a means. And given that Bob would have died anyway, the side-effect is not disproportionate to the good of saving Alice from being a murderer.”

Answer: I think the proportionality condition is not met. Sure, Carl makes Alice not be a murderer. But Alice is still an attempted murderer—which is just as culpable as being an actual murderer—and her malfeasance still causes Bob’s death, so she still has that death on her conscience. Granted, she isn’t a murderer any more (if I am right about Question 1), but the bad of Carl’s accidentally killing Bob seems disproportionate to the relatively minor good achieved here.

It’s interesting when it is the proportionality condition in double effect that ends up being crucial.

Leprechauns and fairies

Why aren’t multiverse—or large universe—hypotheses that imply that there very likely exist fairies and leprechauns dismissed as mere sceptical hypotheses?

More remarks on Aristotelian set theory

If we have an Aristotelian picture of abstracta, we should expect that what mathematical objects exist differs between possible worlds.

For the Aristotelian, abstract objects are abstractions from concrete things. So we shouldn’t expect the same full panoply of sets regardless of what concrete things there are. For instance, suppose that the universe contains exactly three point particles, A, B and C. Then we can immediately abstract from these particle positions distance ratios like AB : BC, AC : AB and AC : BC. These ratios are then represented by real numbers. So we are going to have these real numbers. More sophisticated abstractive processes may well generate other real numbers: for instance, we will have a real number representing the ratio of the height of the triangle drawn from A to the base BC. And given a real number, we might be able to use purely abstract processes to generate further real numbers: given a and b, we may generate a + b and ab, say. But there is no reason to think that these abstract processes will generate the same collection of real numbers regardless of what the three particle positions we start with are.

So, what real numbers exist should vary between possible worlds. But every real number defines a subset of the natural numbers (just write the real number in binary, and let the nth bit decide if n is in the subset or not). If the real numbers vary between possible worlds, so do the subsets of the natural numbers. In particular, we should expect that in different possible worlds, a different set counts as ``the power set’’ of the natural numbers.

Furthermore, what bijections there are between sets will vary between possible worlds. Thus, if we see the question of whether two sets have the same count of members as having the same answer in every world where the two sets exist, we cannot take the standard Cantorian account of the size of a set. Instead, we may want to generate the concept of sameness of size from bijections in different worlds. Thus, we may try to say that two sets A and B are the same size at level 0 provided that there is a bijection between A and B. Then we say that A and B are the same size at level n provided that possibly there is a set C that is the same size as A at level p and the same size as B at level q and n ≥ 1 + p + q. Finally, we say that A and B are the same size simpliciter provided that they are the same size at some finite level. This is complicated, and I haven’t checked under what assumptions it generates a transitive relation (it’s plausibly reflexive and symmetric).

Anyway, the point is this: It is an interesting and not easy philosophical project to work out the set-theoretic consequences of Aristotelianism. This could make a good dissertation.

Elections as hiring decisions

People tend to think of elections for high office as sui generis rather than as what they are: hiring decisions, for a particularly onerous but important job.

Once we see elections for high office as hiring decisions, some things become a bit puzzling.

1. It is often seen as important that a candidate for high office have a strong and sincere personal commitment to a platform. But why? Suppose I hire a lawyer to represent my interests. It might be nice if the lawyer had a strong and sincere personal commitment to the things I want the lawyer to represent me in respect of, but it is not at all necessary. What is needed is that the lawyer further my interests, and do so along rough lines that I may sketch, in a professional and effective way. The lawyer does not need to think that it would be better for the world if I get what I want—she may simply think that it is good to have in place a legal system where almost everyone gets able legal representation for the furtherance of their interests, and that what I want isn’t so bad as to make it immoral for her to represent me. One can even imagine a lawyer who specializes in representing a particular kind of interest without actually sharing that interest, but holding that nonetheless it is important that an interest of that sort should be represented.

2. There is an interest in the personal life of candidates that would be seen as creepy and likely illegal in most other hiring decisions.

Of course, these kinds of things might be appropriate in light of the specific features of high political office. But they shouldn’t be taken for granted.

Voting for Schmitler

Three people are running for election in Germany: Hitler, Schmitler and Bonhoeffer. Bonhoeffer promises just policies but has no chance of being elected. Hitler promises to kill 50% of minorities. Schmitler promises to kill 80% of minorities. You might think that at this point I will raise the difficult question whether it is permissible, all other things being equal, to vote for Hitler. But I won’t raise exactly that question.

Instead, I want to expand on the above scenario in a different way. Schmitler is incompetent and won’t manage to do more than a quarter of the evils he promises, unlike Hitler who is going to exactly what he promises. So, whom should you vote for? Bonhoeffer who has just policies but won’t be elected? Hitler whose policies are less bad than Schmitler’s, but who will do exactly what he promises? Or Schmitler whose policies are much worse than Hitler’s, but who will do much less bad than Hitler?

There is a good utilitarian case for voting for Schmitler. Here’s an argument for this case. Suppose the elections are occurring in the middle of World War II. It seems that a very reasonable thing for Allied spies to do is to ensure that incompetent people run Nazi Germany. One means to that goal is stuffing ballot boxes with votes for Schmitler. And while typically one shouldn’t stuff ballot boxes, this seems to be a case where the stuffing of ballot boxes would be permissible. So, Allied spies, we suppose, are stuffing ballot boxes in favor of Schmitler. Helga is a German resister to Nazism, working for the Allies. She is an excellent prestidigitator and is going to the voting booth with a sleeve full of Schmitler ballots, in order to stuff the box surreptitiously. If the Allied spies are doing the right thing, Helga is doing the right thing.

Now, if Helga can permissibly stuff the ballot box in favor of Schmitler, then she could permissibly do this: put in a vote for Bonhoeffer (or Hitler), then surreptitiously remove that ballot and replace it with a fake ballot in favor of Schmitler. But if that’s permissible for her, then it would be very strange if she wasn’t permitted simply to vote for Schmitler.

So, it seems, it is permissible to vote for Schmitler on the grounds that he is incompetent, despite the fact that his policies are significantly worse than Hitler’s. But if this is permissible, then it would be a fortiori permissible to vote for Schmitler if he promised to kill 5% of minorities, and this would seem to be permissible even if Schmitler were as competent as Hitler. So, we have an argument that it is permissible to vote for a candidate whose policies represent a lesser evil. Of course, one should never endorse an evil, even a lesser one. So, it follows that voting for a candidate is not endorsement of the candidate’s policies.

I am not wholly convinced by the above argument. I feel a certain pull to the strange view that while it would be permissible for Helga to replace her real Bonhoeffer vote with a Schmitler fake ballot, it would not be permissible for her to vote for Schmitler. After all, strange circumstances make for strange conclusions.

Authorless books

I've been imagining a strange scenario. I come across a text that I know for sure was generated by an entirely random process--say, the proverbial monkeys at the typewriter. I look at it. Mirabile dictu, it's coherent and reads just like a literary masterpiece--let's say it's just like something Tolstoy would have written had he written one more novel at the peak of his creative powers.

I think reading this random text could be a disquieting experience. I could read it shallowly, the way one reads some novels for mere entertainment. And in that context, it would work just as well as shallowly reading a real novel. But of course with a masterpiece, one wants to read it more deeply. In doing so, one draws connections between different parts of the texts ("Oh! So that's what that foreshadowed!" or "Ah, so that's why she did that!"), between the content and the mode of expression ("Look at all these short words describing the rapidity of the march"), between what is overtly the text and other texts, ideas, historical events and persons, etc. Drawing such connections, whether explicitly or just as a barely conscious sensation of something there--is a part of the enjoyment of reading a literary masterpiece, when done in moderation. But in our random text, all connections are merely coincidental. Nothing is there on purpose, not even unconsciously. When we read a literary giant like a Plato or a Tolstoy, when we see a compelling connection, we have good reason to think the author meant it to be there, and that sensing the connection is a part of a good reading of the text. But in the random text, there will be no such thing as a good reading or a misreading. And that would have to be disquieting. There is a sense in which we would be inventing all the connections. Reading would be more like creating than like discovering. I suppose death-of-author people think that's already the case with normal novels. But I don't think so. Real connections differ from chance ones.

At the same time, I think that in practice if I were reading this text which is just like a literary masterpiece, I'd end up suspending my disbelief about the author, and just delight in the connections and subtleties, even if they are merely apparent.

But maybe in a world with God there is no true randomness. So maybe the hypothesis of a book where nothing is intended is impossible?

More on life

God is alive, angels are alive, people are alive, dogs are alive, worms are alive and trees are alive. What is it that makes them all be alive, while the Milky Way, the Sun, Etna, a car, a Roomba, and an electron are not? I raised a version of this question recently, and since then have had discussions about it with a number of our graduate students, most extensively with Alli Thornton and Hilary Yancey, to whom I am very grateful.

We could say that they are all alive in an analogical sense. But that doesn’t solve the problem, but simply puts a constraint on the shape of a solution. For to solve the problem, we still have to say something about how this particular analogy works.

Here is the best answer I have right now, but it still has some difficulties I will discuss:

• A living thing is one that can act in pursuit of its own ends.

This indeed covers God, angels, people, dogs, worms and trees. Moreover, it produces a gradation of life in respect of the degree and quality with which the thing can act in pursuit of its own ends and the degree of ownership the thing has over its ends. For instance, God is omnipotent and perfectly rational, and he is his own end, so he is most fully alive. All the living creatures, on the other hand, have ultimate ends imposed on them by their nature, to the realization of which end their activity are ordered. However, angels and people additionally make a rational choice of ways to realize their ultimate ends, adopting which ways involves setting themselves intermediate ends. Higher non-human animals like dogs do something that approximates this. Moreover, angels, people and dogs have a wide variety of ways of pursuing that end. On the other hand, trees only pursue a limited variety of ends with a limited variety of means.

A bonus of this definition is that we get the conclusion, which seems intuitively correct, that anything that thinks is alive. For thinking is an end-directed activity—it is directed at action and/or truth. So if we ever make an artificial intelligence system that really thinks, it will be alive.

The Milky Way, the Sun, Mount Etna, a car and a Roomba do not pursue their own ends, I think, if only because they are not substances, and only substances own their ends.

But electrons… This is what troubles me. I think that the fundamental constituents of physical reality, be they particles or fields, are substances. And I think all substances have a teleology, and hence have an end. The distinction I would like to be able to make, however, is between activity in pursuit of an end and teleological activity more generally. Electrons in their characteristic activity are acting teleologically. But their action is not in pursuit of an end. Rather their end is simply to engage in this very activity and nothing more. The activity is teleological, but it does not pursue a telos.

But what if it turns out that electrons do genuinely act in pursuit of an end? Then, perhaps, we will have learned that electrons are a very primitive form of life.

What about God, given divine simplicity, though? God = God’s telos = God’s activity. Well, I think that even if God’s activity is identical with God’s telos, one can make a conceptual distinction that allows one to say that God acts for the sake of that telos, a distinction that perhaps is not there in the case of the electron.

As you can see, I am still not very happy with the account. But it’s the best I have right now.

If naturalism is true, there is an infinite afterlife

1. Deontology is true.
2. A finite being could not have the kind of dignity that deontology ascribes to human beings.
3. So, human beings are infinite. (1-2)
4. If human beings are infinite, they are infinite synchronically or diachronically.
5. If naturalism is true, human beings are finite synchronically.
6. If there is no infinite afterlife, human beings are finite diachronically.
7. So, either naturalism is not true or there is an infinite afterlife. (3-6)
8. So, if naturalism is true, there is an infinite afterlife. (this is a material conditional following from 7)

Of course, there are also arguments that if naturalism is true, there is no afterlife, and if these are sound, then together we get an argument that naturalism is not true.

Eternal life and the crucible of character theodicy

Consider the crucible of character theodicy, that we are permitted by God to meet with great evils in order to form a character with virtues like courage and sacrificial love whose significant exercise requires significant evils.

I take it that it’s clear that forming such a character is worthwhile. But there are at least three problems with this theodicy:

1. While such character formation is valuable, is it valuable enough to justify our suffering great evils? Wouldn’t it have been better if God just gave us the virtues directly, rather than having us pay a great price?

2. Even if it is valuable enough to justify our suffering great evils, wouldn’t it be better if we suffered fewer or lesser ones?

3. What about those who suffer and develop a vicious character?

I think these three problems can be overcome if we think about heavenly life as an infinite value multiplier.

Ad 1: There is clearly some additional value to having virtues that were formed through significantly free exercises of them rather than having had these virtues imposed on one. In heaven, on infinitely many days one has and enjoys the value of having virtues. But if one has formed these virtues through significantly free exercise, then on infinitely many days one also has and enjoys the additional value of having virtues that were thus formed. That’s an infinite additional increment. So as long as the disvalue of the sufferings in this life was finite—which surely it was—it’s worth it.

Ad 2: The greater the sufferings that one endured courageously and the greater the sacrifices one made in love, the more fully one owns the resulting courage and love. For in more extreme exercises of these virtues, one has a greater opportunity to abandon the path of virtue, and one’s presence on that path is more truly one’s own. And this deeper ownership over one’s virtue—bearing in mind, of course, that all one has is a participation of God, and that grace is deeply involved—adds an additional value of virtue-ownership throughout an infinite number of future days. Hence, it adds an infinite amount of value, which is surely worth it.

Ad 3: This is probably the most serious worry. Start with this thought. God is choosing whether to snatch Judas up to heaven in the first moment of his existence, imposing on him a perfectly virtuous character, or to give Judas the opportunity to freely develop and own that character. A toy model for this an extended utility calculation. On the first option, we have an expected utility of

• V(eternal unowned virtue),

where V is value. On the second option, we have an expected utility of

• pV(eternal owned virtue) + (1 − p)V(Judas chooses vice)

where p is the probability that Judas would come through the crucible well. (Of course, this line of thought requires rejecting theological compatibilism and Molinism.) Here, V(eternal unowned virtue) and V(eternal owned virtue) are each infinite and positive. Plausibly, V(Judas chooses vice) is negative. Is it infinite? That’s not clear. One might think that on orthodox Christian views of hell, it is both negative and infinite. But that need not be the case. It could be that the suffering and vice in hell actually decreases from day to day, so that the total amount of suffering and vice over eternity is actually finite (think of how 1 + 1/2 + 1/4 + ... = 2).

If V(Judas chooses vice), the argument still isn’t over, but I will assume that V(Judas chooses vice) is finite—we could just build that into the theodicy. In that case, we can basically neglect V(Judas chooses vice)—when the other quantities are infinite, a finite subtraction is only going to be a tie-breaker.

So now the question is whether V(eternal unowned virtue) is bigger than or equal to pV(eternal owned virtue). And here it seems very reasonable simply to make a sceptical theist move. We don’t know what was Judas’ probability of coming through the crucible well. We don’t know exactly how V(eternal owned virtue) compares to V(eternal virtue). It could be that a day with owned value is three times as valuable as a day with unowned virtue. If so, then as long as p > 1/3, God’s giving Judas the opportunity for freely choosing virtue was worthwhile.

There are many objections, of course, that one can make. Here’s one that particularly comes to my mind: Wouldn’t it be better for God to first give people the opportunity to freely choose a virtuous character, but then if they refuse to do so, to impose that character on them? After all, at least some infants go to heaven after death. But they haven’t developed a virtuous character through the described kind of crucible. And so it seems that God imposes on them a virtuous character.

There are two things I’m inclined to say to this. First, there is a relevant difference between the case of imposing virtue on an infant and imposing virtue on someone who has chosen against virtue. Second, those who choose virtue own their virtue more fully if they had the possibility of not having that virtue at all.

Three levels of sex/gender

The biological understanding of male and female is something like this. Some species reproduce sexually. Some species that reproduce sexually exhibit a consistent difference in size between the two gametes that come together in sexual reproduction. In those species, the producer of the larger gamete is called “female” and the producer of the smaller gamete is called “male”. We can thus draw a distinction between a species having sexes, namely having respective producers of two different kinds of gametes, each of which is needed for sexual reproduction, and the species having male and female sexes.

Let me speak vaguely but heuristically. Human reproduction has a deep ethical and theological significance because it produces persons. Moreover, humans normally reproduce sexually (the exception of course being twinning). So it’s unsurprising if the existence of two sexes among humans has intrinsic ethical and theological significance. But the difference between male and female seems to have no intrinsic ethical or theological significance. It matters that there are two reproductive kinds, but that one of the two kinds produces a larger gamete than the other has no intrinsic ethical or theological significance.

But of course even though what defines the difference between male and female humans is the difference in gamete size, the actual differences between male and female humans are not in fact limited to differences of gamete size. Those humans that produce smaller gametes produce more of them, while those humans that produce larger gametes produce fewer of them and gestate offspring. Humans have “primary sex characteristics” that support differing ways of reproductive functioning.

Here is a thought experiment. Imagine earth* where there are humans*. To a cursory external examination, humans* live, look and behave just like humans, and have the same kind of sexual differentiation. One sex produces lots of gametes and the other relatively few. The sex that produces fewer gametes gestates offspring for nine months, has mammary-type glands that nourish offspring after gestation, is a little smaller on average, etc. But on earth*, it also turns out that the the sex that produces relatively few gametes produces the smaller gametes. (There may be evolutionary reasons why this is unlikely. But unlikely is not impossible.) Thus, on earth* male humans* fill the same biological roles as female humans do on earth, except at a near-microscopic level where the sizes of gametes become visible.

Now overlay on this the social level. This could go in multiple ways. It is easiest to imagine that on earth*, male humans* have the same social positions, and suffer from the same sorts of discrimination, as female humans do on earth. But it could in principle be reversed: it could be that the social position of male and female humans* is like that of male and female humans, respectively. Or it could be nullified: there could be no significant differences in social position.

This suggests that there are three levels to sex/gender:

• The definitionally fundamental distinction between male and female in terms of gamete size.

• Other biological differences—particularly with respect to reproductive functioning.

• The social distinctions.

The first two tend to be lumped together as “sex” or “biological sex”, while the last gets called “gender”. But there really are three distinct levels. We might roughly call them: “biological gametic sex”, “biological functional sex” and “social gender”. Thus, among humans*, the connection between biological functional sex and biological gametic sex is the reverse of how it is among humans. So we now have three different senses of terms like “man”, “woman”, “male” and “female”.

Some paradoxes of reference

Liar-like:

• one plus the biggest integer that can be expressed in English in fewer than fifty words
• one; two; three; one plus the biggest integer mentioned in this list
• one; two; three; one plus the last integer mentioned in this list
• one plus the last integer mentioned in this list; two; three; one plus the first integer mentioned in this list
• one plus this integer

Truthteller-like:

• one; two; three; the biggest integer mentioned in this list
• one; two; three; the last integer mentioned in this list
• the last integer mentioned in this list; two; three; the first integer mentioned in this list
• this integer
• the square of this integer

Heaven

One of the things that most strengthens my faith is compelling accounts of heavenly happiness. I think there are at least two reasons why such accounts tend to strengthen faith. First, a compelling account of heavenly happiness rebuts the "sounds like fairy tale" objection to Christian faith. Second, a compelling account of heavenly happiness resonates with us in a way that gives us evidence that we are meant for a different kind of life than the one we have hear. I wish there were more sermons on heaven. And we philosophers also should work more on giving good accounts of what heaven might be like.

Disjunctive victimization?

Alice, Bob and Carl are suffering from a deadly disease. Alice possesses one dose of a medication necessary and sufficient to cure the disease. She has four relevant options. First there are:

1. Use the medication on herself.
2. Use the medication on Bob.
3. Use the medication on Carl.

Option 1 is permissible. Options 2 and 3 are supererogatory. But what Alice actually does is:

4. Destroy the medication.

Alice clearly did something wicked by failing to use the medication to save a life. But how do we describe this wicked deed?

It seems that Alice’s action was a fatal negligence of a duty towards herself, and a fatal negligence of a duty towards Bob and a fatal negligence of a duty towards Carl. But that makes it sound like three counts of fatal negligence, which is triple-counting the wrongful act.

I suppose what we can say is something like this: Alice neglected to use the medication to save a life. Whom did she act against? Maybe each of: herself, Bob and Carl. But we shouldn’t look at the action as the violation of three duties, but only of one duty, to use the medication to save herself, Bob or Carl. So she violated a single duty, to the tune of a single life, but that single duty was one she owed to three people.

Question 1: Does it follow that one can have a duty to a group which does not reduce to a duty to each member? For Alice surely doesn’t owe Bob that she save herself, Bob or Carl, and she doesn’t owe Bob that she save Bob, since she can permissibly save herself or Carl.

Answer: I don’t know. Maybe we can say:

• Alice owes herself that if she doesn’t use the medication to save Bob or Carl, she use it to save herself
• Alice owes Bob that if she doesn’t use the medication to save herself or Carl, she use it to save Bob
• Alice owes Carl that if she doesn’t use the medication to save herself or Bob, she use it to save Carl.

And so Alice wrongs each of herself, Bob and Carl. A problem with this solution, however, is that it seems to triple counting Alice’s wrongdoing, by making it seem like she fatally wronged each of three people—but she is only responsible for a single death. Maybe, though, we can say that the duty to the three reduces to the three individual duties, but that the culpabilities don’t sum?

Question 2: Does the case provide an argument that one can wrong oneself? My above description of the case as one one where Alice owes it to herself, Bob and Carl that she save herself, Bob or Carl presupposes duties to herself. What can someone who thinks there are no duties to self say?

Answer: I don’t know. Maybe she can say: Alice owes Bob and Carl that she save herself, Bob or Carl. But it would be a little weird to think that by saving herself, Alice would be fulfilling a duty to Bob and Carl.

Final remarks: I am far from clear how to morally describe the case. I think the neatest description is one where a group is non-reducibly victimized, and where there are duties to self. But that may not be the only admissible description.

Instantaneous Newtonian gravitational causation at a distance?

It’s widely thought that Newtonian gravity, when causally interpreted, involves instantaneous causation at a distance. But I think this is technically not right.

Suppose we have two masses m₁ and m₂ with distance r apart at time t₁. The location of m₂ at t₁ causes m₁ to accelerate at t₁ towards m₂ of magnitude Gm₂/r². And this sure looks like instantaneous causation at a distance.

But this isn’t an instance of instantaneous causation. For facts about what m₁’s acceleration is at t₁ are not facts about how the mass is instantaneously at t₁, but facts about how the mass is at t₁ and at times shortly before and after t₁: acceleration is the rate of change of velocity over time. Suppose that a poison ingested at t₁ caused Smith to be dead at all subsequent times. That wouldn’t be a case of instantaneous causation, even though we could say: “The poison caused t₁ to be the last moment of Smith’s life.” For the statement that t₁ is the last moment of Smith’s life isn’t a statement about what the world is instantaneously like at t₁, but is a conjunctive statement that at t₁ he’s alive (that part isn’t caused by the poison) and that at times after t₁ he’s dead (that part is caused by the poison, but not instantaneously). Similarly, m₁’s velocity (and position) at times after t₁ is caused by m₂’s location at t₁, but m₁’s velocity (or position) at t₁ itself is inot.

Let’s call cases where a cause at t₁ causes an effect at interval of times starting at, but not including, t₁ a case of almost instantaneous causation. In the gravitational case, what I have described so far is only almost instantaneous causation. Of course, people balking at instantaneous action at a distance are apt to balk at almost instantaneous action at a distance, but the two are different.

The above is pretty much the whole story about instantaneous Newtonian causation if one is not a realist about forces. But if one is a realist about forces, then things will be a bit more complicated. For m₂’s location at t₁ causes a force on m₁ at t₁, which complicates the causal story. On the bare story above, we had m₂’s location causing an acceleration of m₁. When we add realism about forces, we have an intermediate step: m₂’s location causes a force on m₁, which force then causes an acceleration of m₁. (There might even be further complications depending on the details of the realism about forces: we may have component forces causing a net force.) Now, when the force-at-t₁ causes an acceleration-at-t₁, this is, for the reasons given above, a case of almost instantaneous causation. But the causing of the force-at-t₁ by the location-at-t₁ of m₂ is a case of genuinely instantaneous causation.

But is it a case of causation at a distance? It seems to be: after all, the best candidate for where the force on m₁ is located is that it is located where m₁ is, namely at distance r from m₂. (There are two less plausible candidates: the force acting on m₁ is located at m₂, and almost instantaneously pulls on m₁; or it’s bilocated between the two locations; in any case, those candidates won’t improve the case for instantaneous action at a distance.) But here is another problem. The force on m₁ is not produced by m₂. It is produced by m₁ and m₂ together. After all, the Newtonian force law is Gm₁m₂/r². (It is only when we divide the force by m₁ to get the acceleration that m₁ disappears.) Rather than m₂ pulling on m₁, we have m₁ and m₂ pulling each other together. Thus, m₂ instantaneously partially causes the force on m₁ at a distance. But the full causation, where m₁ and m₂ cause the force on m₁, is not causation at a distance, because m₂ is at the location of that force.

In summary, the common thought that Newtonian gravitation involves instantaneous causation at a distance is wrong:

• If forces are admitted as genuine causal intermediates (“realism about forces”), then we have almost instantaneous causation of acceleration by force (moreover, not at a distance), and instantaneous partial causation of force at a distance.

• Absent force realism, we have almost instantaneous causation at a distance.

Life

Accounts of biological life characterize it by lists of features such as “reproduction, metabolism, functional organization, growth, responsiveness to the environment, movement, and short- and long-term adaptations” (SEP s.v. life). But Jewish, Christian and Muslim theists have reason to worry about such accounts of life in the light of the fact that our scriptures present God as the paradigm of a living being.

Here are some options:

Option 1: Modify one’s theology to make God fit with something pretty close to one of the biological accounts of life. Mormonism is a fairly radical example of this. A more moderate version might be some version of process theology, though one may need to jettison some features, like metabolism and growth.

Option 2: Trinitarians have this option available: The Son proceeds from the Father and the Spirit from the Father and/through the Son and these processions could count as reproduction. Moreover, because Trinitarian processions multiply persons but the resulting persons are one God, Trinitarian reproduction has an internality that might count as a kind of growth—not that the divine essence grows, but that the number of persons of the one God grows. This solution would have the important consequence that the Old Testament texts that present God as living are proto-Trinitarian. Obviously, this solution is unlikely to be attractive to Jews and Muslims, though there may be some Kabbalistic analogue that might appeal to some Jews. Moreover, unless one adopts some version of process theology, this solution still requires one to drop a number of the features in traditional accounts of biological life, such as metabolism, movement and adaptation.

Option 3: Replace the biological accounts of life with something radically different which makes God a paradigm instance of life. Here are three such possibilities for characterizing life:

A. Living things are ones that have some mental property like consciousness or purposefulness.

B. Living things have teleology.

C. Living things are ones that have a well-being, that are capable of being well.

Suboption A is pretty radical: it requires either saying that plants have a mental life or that plants aren’t alive. I think it’s not that crazy to say that plants have something like mental life. Maybe they are aware of their environment in a way that goes robustly beyond the mere data processing of a digital thermometer. And it seems plausible that one can ascribe a certain kind of purposefulness to plant processes.

Suboption B is pretty close to the purposefulness variant of suboption A, but teleology is a more general concept than mental purposefulness. For me, the main difficulty with suboption B is that I think all substances have teleology. And I don’t want to extend life to elementary particles. But those who do not think that teleology extends to all substances might like Suboption B.

Suboption C is, of course, related to Suboption B. I have the same worry about C as about B: I think all substances that have teleology have a well-being. Elementary particles have well-being—the only difference between them and organic substances is that as far as we can tell, elementary particles are always well. This is of course very controversial, and those who do not accept it may like C.

There are, no doubt, other options and suboptions. I am attracted to Options 2 and 3A.

Priors, justification and rationalism

The rationalism of Leibniz and Spinoza worked like this: We figure out fundamental necessary metaphysical principles, and these principles determine everything else of necessity (with some qualifications on the Leibniz side as to the type of necessity).

But another rationalism is possible: We figure out fundamental necessary metaphysical principles, and these principles determine the basic probabilistic structure of reality. In Bayesian terms, the fundamental metaphysical principles yield the prior probabilities. A version of this was Descartes’ project in the Meditations.

And there is reason to engage in this probabilistic rationalist project. We cannot get out of the need to have something like prior probabilities. Moreover, priors need epistemic justification. Consider an empirical claim p that we assign a high enough credence for belief to, say 0.99, on the basis of total evidence e. Thus, P(p|e)=0.99. It follows by the axioms of probability that P(p ∨ ¬e)≥0.99. Hence we have a high enough prior credence for belief in p ∨ ¬e. Surely assigning a credence of 0.99 to something requires epistemic justification. Moreover, surely (though people who don’t like closure arguments may not like it) if we have posterior justification when we believe p, we have posterior justification when we believe the obviously entailed claim p ∨ ¬e. But this justification did not come from e. For P(p ∨ ¬e|e)=P(p|e)=0.99 and we have seen that P(p ∨ ¬e)≥0.99, so e is not evidence for p ∨ ¬e (in face, typically e will be evidence against this disjunction). Since e is our total evidence, the justification had to be there in the first place.

Thus we need epistemic justification for our priors. The priors encode genuine information about our world, information that we are justified in possessing. Where do we justifiedly get this information from? We don’t get it through logic, pace logical probability accounts. Metaphysics is one potential answer to this question and exploring this answer gives us good reason to engage in the probabilistic rationalist project. Another option is that the priors are a kind of innate knowledge built into our nature—my Aristotelian Bayesianism is a version of this.

Multiple levels of multiple realizability

We could have sophisticated beings who reason about the world via numerical Bayesian credences. But we could also have sophisticated beings who reason in some other way—indeed, we are such beings. And there is one sophisticated being who reasons about the world via omniscience. This suggests that reasoning and agency are multiply realizable at multiple levels, including:

1. brain/mind architecture

2. algorithms implementing general reasoning and representation strategy

3. general reasoning and representation strategy.

Each level is an abstraction from the previous. So now we have a very deep question: Is there a fourth level that abstracts from the third, to get the concept of reasoning as such? Or are the various general reasoning and representation strategies unified analogically, say by similarity to some primary case? And if so, what is the primary case? Omniscience? Logical omniscience plus numerical Bayesianism?

A pessimistic thought about epistemic rationality

The more I think about relatives of the Doomsday Argument and my insect argument, the more I become suspicious of both the self-sampling and the competing self-indication assumptions.

Both assumptions require one to assign non-zero probabilities to metaphysically impossible hypotheses about where I am located in spacetime, such as the hypothesis that I was born in the 19th century, the hypothesis that I am a non-human or the hypothesis that it is now a Wednesday. It is usual in the discussion of these hypotheses not to worry much about such issues of metaphysical impossibility. The thought is that we are doing epistemology, not metaphysics.

Moreover, it seems reasonable to assign a credence 1/10 to the proposition that the hundredth digit of π is 3 when one hasn’t calculated that digit, even though one knows that the proposition is either necessarily true or necessarily false.

Perhaps my Aristotelian meta-epistemology can help, though. What if the right way to assign credences is not some abstract “rational” way to do so, but rather the human way—the way that our human nature calls us on us to? Our human nature specifies how we should function within a certain broad range of conditions. But outside of that range, human nature may not specify how we should function. Our nature specifies normal human locomation in earth gravity, but specifies nothing about what normal human locomation out in space should be like. Likewise, our nature specifies normal human credential behavior in many circumstances, but need not specify human credential behavior in outlandish cases like Doomsday.

Of course this raises the very hard question of which cases are within the range of cases where our nature specifies credential behavior and which are not.

Super-simple fractal terrain generator

Here's a very simple algorithm for generating random terrains: Start with a equilateral triangle in the x-y plane. Add a uniformly and symmetrically distributed random z-offset to each vertex. Bisect each resulting edge and add a random z-offset at the newly added point, half of the magnitude of the random offsets earlier. Repeat several times.

The algorithm is no doubt known, but some quick searches for terrain generation algorithms didn't turn it up, so I am posting for people's convenience.

There are some artifacts at internal triangle boundaries that better algorithms presumably don't have, but the algorithm is super simple to implement, and because it is based on triangles it directly makes a triangular mesh. Here is some code which does this, starting with a hexagon divided into six equilateral triangles, and putting out an STL file.

Are the priors of subjective Bayesianism subjective?

Visual and auditory perception are not subject to rational evaluation. Nobody perceives visually or auditorily in a rational or irrational way. Nonetheless, perception is subject to evaluation. One’s perceptual faculties could function superlatively, adequately or inadequately. The mere fact that they are not subject to rational evaluation does not imply subjectivism about their functioning.

But now consider a Bayesian view on which:

1. Our priors are not subject to rational evaluation.

That view is known in the literature as “subjective Bayesianism”. But if we take seriously the lesson from perception, we should be sceptical of the inference from (1) to:

2. Our priors are merely subjective.

I have to confess to not taking this point seriously in the past, having been misled by the phrase “subjective Bayesianism” and by things I heard from subjective Bayesians.

What might a theory look like on which our priors are subject to evaluation but not rational evaluation? We could take our priors to be a kind of “probabilistic perception” of patterns in the world, a perception that is genetically and/or socially mediated. Such perceptions can be better or worse, just as the person who is looking at a horse and their visual system classifies it as 95% likely to be cat and 5% likely to be a horse is doing less well perceptually than one whose system makes the opposite classification. For instance, someone who has a prior close to 1 for the law of gravitation being an inverse 3.00001th power law is doing less well than someone who has a moderately high prior for it it being an inverse cube law and a moderately high prior for it being an inverse square law.

But if we take the perception analogy seriously, we get this question: What are we “perceiving” with our priors? Maybe something like facts about the sorts of laws worlds like ours have?

Probabilistic perception

We could imagine critters whose perceptual system works as follows: When they have an object in their visual field, instead of the perceptual system delivering the presence of a dog, it delivers something like:

• dog:0.93, coyote:0.03, wolf:0.03, deer:0.01.

There are probably many interesting questions to ask about critters with a perceptual system like that. But I want to briefly muse about three.

Question 1: Can we redescribe the perceptual system of these critters so that at base what we have are just attitudes to propositions or properties pairs or something like that?

Answer: I am not sure. Some options fail. For instance, while it may be true that the critter in my example is having a disjunctive perception of a dog-or-coyote-or-wolf, that doesn’t capture all the information in its perceptual system—it doesn’t capture the much greater probability of its being a dog.

Alternately, one could say that the critter has four perceptions of different strengths: a 0.93 strength perception as of a dog, a 0.03 strength perception as of a coyote, a 0.03 strength perception as of a wolf and a 0.01 strength perception as of a deer. But that doesn’t quite capture what’s going on, at least not if we read the story as I intended it. The critter takes dog, coyote, wolf and deer to be alternative hypotheses for what is in front of it, not to be four different perceptions. The story that breaks up the perception into four perceptions of different strengths fails to distinguish the story I intend from a story where the animal might be all four (it’s only a posteriori that we know there are no dog-coyote-wolf-deer).

Maybe we could say that the critter’s perceptual system also delivers something more complicated:

• only-dog:0.93, only-coyote:0.03, only-wolf:0.03, only-deer:0.01.

That will get out of the alternativeness worry, but I am sceptical that it needs to be like that. One could just see the four options, and not see that the probabilities involved force them to be exclusive (because the probabilities add up to one). This is even more plausible if the probabilities are qualitative or interval-based.

Nor will it do to say that one perceives that there is a 0.93 probability of a dog, a 0.03 probability of a coyote, and so on. For these probabilities are not objective facts out there. They are, I suppose, measures of what credence the critter should have in each hypothesis if there is no further data available. We need not suppose that the critter has the degree of self-reflectiveness that would be needed to perceive these measures of hypothetical credence as such.

So maybe a reduction to more familiar perception stories is possible, but I think there is some reason to be sceptical.

Question 2: What is it for this perceptual state to be veridical?

Answer: A necessary condition, of course, would have to be that what is present is a dog, coyote, wolf or deer. But there is room for much Gettiering. Maybe it’s a wolf dressed up as a sheep dressed up as a wolf. Then the perception isn’t in the right way, and we don’t have veridicality. But what if it’s a dog that recently went to the pet salon and was made up to look more wolf-like? Then maybe it’s veridical. Maybe. I just don’t know.

I have a suspicion that once we have such probabilistic deliverances of perception, the in-the-right-way problem of characterizing veridicality not only becomes epistemically intractable—it may already be that in standard theories of perception—but the whole concept of veridicality, apart from the necessary condition that one of the alternative hypotheses be true, may break down.

Question 3: Are we always such critters? We could, after all, take the ordinary perception of a dog to be just a limiting case like “dog:0.999999, something weird:0.000001” or even “dog:1”. Should we do that in every case?

Answer: Phenomenologically, the answer seems negative. But phenomenology can mislead about such things. But as long as it’s a live hypothesis that we might be such critters, we may need to be cautious about claims like that perception is a propositional attitude (see Question 1) or that there is a viable concept of veridicality (see Question 2).

And the apparent possibility of critters whose perception always works like this should make us cautious as to the kinds of claims we make in epistemology.

Diversity of goods, heaven and theodicy

Alice was tortured for weeks, but she remained loyal to the true and the good. Alice’s moral victory over her torturers was of great value. Let’s take it for granted that this situation represents something that is on balance a great good for Alice (and for the world). But suppose that God was choosing between Alice’s moral victory, on the one hand, and Alice doing something else that was virtuous, worthwhile but not painful—perhaps with great singlemindedness throwing herself into producing a great work of art. One might well make the judgment that while Alice’s moral victory was a great good, given the horrendous cost to Alice, it would on balance have been better for God to have steered her life away from the suffering and towards a good that comes without much pain.

I am sceptical of this judgment, but for the sake of the argument let’s grant it. Nonetheless, I think the assessment should change if one adds to the above a story about an infinite heavenly afterlife. Here is why. The judgment was that while there was great value in Alice’s moral victory, it would have been good to substitute a painless but valuable achievement for it.

But what if, instead, the question was this:

• Is it better for Alice to have the painful moral victory plus a hundred painless but valuable achievements, or just the hundred painless but valuable achievements?

If we grant the initial judgment that Alice’s moral victory was of great value, then we need to say that the first option is on balance better. The life with the painful victory and hundred painless achievements not only exhibits the additional good of the painful victory, but also exhibits the higher-order value of a diversity of types of goods.

Now ask this question:

• Is it better for Alice to have the painful moral victory plus an infinite number of painless but valuable achievements, or just the infinite number of painless but valuable achievements?

It sure seems like the first option is the better one. But now when we ask the initial question, whether God shouldn’t have given Alice a painless achievement instead, in the context of a theory of God that includes an infinite happy afterlife, that is in fact what the question comes to. The additional painless achievement would not add much to the infinitely many painless achievements in heaven. But the painful achievement adds something different in kind.

And when we add to the story that in the afterlife Alice (as well as her friends, likely many in number) will be able to enjoy, infinitely many times, the memory of her moral victory, without the memory of her suffering being itself a source of suffering (for, I take it, there is no suffering in heaven), it sure seems worth it.

But what if instead Alice cracked under torture, thereby losing the good? Well, now how well the story works depends on other questions. If we have Molinism, then it doesn’t work very well: given Molinism, God can foresee that Alice would crack, and so the value that there would be in her victory is irrelevant. But given simple foreknowledge (or open theism, but that’s a heretical option) God’s decision whether to put Alice in that situation can only be based on chances. For all we know, the chance of Alice’s cracking under torture, given the grace that God gave her, was not very high, and so the significant chance of the great goods of having a diversity in the types of goods one experiences and of having an infinite number of reminiscences of that diversity was worthwhile. Or, at least, we have little reason to think it wasn’t.

Internalism, externalism and Bayesianism

Maybe a Bayesian should be a hybrid of an internalist and an externalist about justification. The internalist aspect would come from correct updating of credences on evidence, internalistically conceived of. The externalist aspect would come from the priors, which need to be well adapted to one's epistemic environment in such a way as to lead reasonably quickly to truth in our world and maybe also in a range of nearby worlds.

This seems a natural way to think about the internalist and externalist question by means of the analogy of designing a Bayesian artificial intelligence system. The programmer puts in the priors. The system is not "responsible" for them in any way (scare quotes, since I don't think computers are responsible, justified, etc--but something analogous to these properties will be there)--it is the programmer who is responsible. Nonetheless, if the priors are bad, the outputs will not be "justified". The system then computes--that is what it is "responsible" for. It seems natural to think of the parts that the programmer is responsible for as the externalist moment in "justification" and the parts that the system is "responsible" for as the internalist moment. And if we are Bayesian reasoners, then we should be able to say the same thing, minus the scare quotes, and with the programmer replaced by God and/or natural selection and/or human nature.