Relativity of simultaneity

I've been thinking about Einstein's nice argument for the relativity of simultaneity in his popular book. The argument starts with the assumption that the speed of light is the same in every inertial reference frame, and uses this to construct a method for determining whether two events are simultaneous. Basically, this method involves having an inertial observer spatially equidistant between the two events checking whether light reaches her simultaneously from the two events. Given the constancy of the speed of light and the equidistance assumption, it seems to follow that the two events are simultaneous if and only if light reaches the observer simultaneously from the two events. And then Einstein gives a very nice argument that applying this method gives different answers depending on the osberve, and concludes that simultaneity is relative to the reference frame.

But there is something that has been worrying me conceptually about Einstein's account of simultaneity. That account takes for granted that we know what it means for the observer to observe two events simultaneously. But isn't the task to define simultaneity?

I guess not. Einstein seems to presupposing that we already have the notion of simultaneity of events at the location of an observer. Moreover, the details of Einstein's argument assume this principle which I think he doesn't discuss:

• Two events befalling the same observer occur simultaneously in the reference frame of the observer if and only if they occur at the same point in spacetime.

(Einstein also tacitly makes the simplifying assumption that observers are point-sized. I won't worry about that assumption in this post.) I am a little troubled by this principle. It's not clear that it's conceptually necessary (might we not think it's violated in cases of time travel?). Still, maybe the best way to take Einstein's account of simultaneity in the book is this. First, we define simultaneity for events befalling the observer who defines a reference frame by requiring sameness of spacetime location. Second, we use this and the equidistant-observer thought experiment to define simultaneity for events not both located at the observer. Third, we show that by this two-part definition of simultaneity, simultaneity is frame-relative.

Is music a sound?

It seems that music is a sound, and sound is constituted by vibrations of a surrounding medium, i.e., typically air. But you can listen to a musical piece through a bone conduction headset. In that case, you're not listening to vibrations of a surrounding medium. Moreover, we could suppose that the performer is producing electronic music live, which is directly piped to the audience's bone conduction headsets without any speakers anywhere, so it's never in the air (except accidentally). Assuming that the bone conduction headsets produce the same experienced quality as listening to the music the normal way, it seems that the audience wouldn't be losing out on anything musically relevant. Yet, if music is a kind of sound, and sound is vibrations of a surrounding medium, then we have a paradox:

1. There is no music there.
2. But the audience isn't missing out on anything musically relevant.
   
3. So, music need not be musically relevant!

Perhaps, though, music doesn't require vibrations of a surrounding medium. The vibrations of bones might be sufficient to qualify as music. I am not sure, however, whether in the concert that I have imagined the audience counts as hearing vibrations of their bones. Yes, their bones vibrate, but the content of the experience isn't the vibration of bones. Rather, it sounds like sound coming from outside them, so the content is external sound, but in my story that's absent, replaced by a mere illusion of sound.

In any case, we can modify the story. Suppose the piece is performed electronically, and never generates the relevant vibrations. Instead, it is directly piped to the performer's and audience's brains' auditory centers. It seems that musically nothing is lost, even though now there is definitely no sound at all.

The conclusion that music needn't be musically relevant is absurd. So we have to deny the claim that music is a sound. What is it then? A sequence of experiences? Then there is no music when the performer and audience are deaf. Maybe that's a bullet we should bite?

Maybe music--both the music composed by a composer and the music performed by a performer (who may be in part or whole a composer, as in cases of improvisation)--should be seen as an abstract sound type. The composer and performer discover music but don't create it. In order to grasp an abstract sound type, it is not needed that one hear an instance of it, but only that one have an experience as of hearing an instance of it. The performer, thus, causes the audience to have experiences as of hearing instances of it. Those experiences are neither sound nor music. We can then, by extension, call an instance of the abstract sound type--i.e., a concrete sound--"music". But music in this sense is not musically relevant except as a vehicle for music in the Platonic sense.

(In case it's relevant, I should note that I'm largely tone deaf, and I do not speak from experience.)

(One can mount another argument against the thesis that music is a sound on the basis of Cage's 4'33". But if 4'33" is music, we could still say it normally involves sound, in that one could have a more nuanced theory on which sound isn't the vibrations, but a token pattern of vibrations. And no-vibrations counts a pattern of vibrations.)

A reason why God might not give second chances

Jones definitively rejects God in this life. He dies. Should God give Jones a second chance at salvation? While an endless--or just extremely long--sequence of second chances might damage Jones' freedom to decide his ultimate destiny, a single second chance seems to be clearly a good thing.

Not necessarily! By giving Jones a second chance to choose God, God would also be giving Jones a chance to reject God all over again. But it is much worse to do wrong than to have bad things merely happen to one, at least when the wrong and the bad are proportionate. And rejecting God is among the worst of all wrongs--maybe even the worst of all wrongs. So there is definitely a risk of further gravely harming Jones by giving him a second chance.

This risk was already present when God gave Jones a first choice for salvation. But once one has done something terrible, doing it again is easier. If Jones has once rejected God's overtures, rejecting them again will be more probable, other things being equal. So, normally, the risk increases. Granted, God could decrease this risk to the level of the first-chance risk by changing Jones' character, but in doing so, God would be overriding Jones' freedom to decide on his character.

None of these considerations show that God shouldn't give a second chance. God could override character or take the risk of letting Jones reject him all over again. But what the considerations do show is that God could be acting reasonably and lovingly towards Jones in not giving Jones a second chance.

This argument depends on theologically incompatibilist simple foreknowledge or open theism: it doesn't work given theological compatibilism or Molinist.

"The Word of God" and infallibility

A couple of days ago, I was reading an article whose author first committed to the Bible being "the Word of God" and then a page later said that the Bible is not infallible in any way. I found this very puzzling. It seems:

• If an assertion of p is the word of x, then either: (a) p is true, or (b) x is mistaken about p, or (c) x is lying about p.

But God is essentially omniscient, so he can't be mistaken about anything. And surely it's foundational of our relationship with God that "God is not a man that he should lie" (Numbers 23:19). So every assertion in the Bible is true if the Bible is God's word.

Now, granted, there may be a bit of a gap between saying that every assertion in the Bible is true and saying that the Bible is infallible. One might note that there are speech acts other than assertions in the Bible, and infallibility for these speech acts comes to something else. For instance, there are commands in the Bible. I don't know what infallibility would come to in the case of a command, but it is plausible that whatever exactly infallibility would come to in the case of a command, a command from God would have that feature.

I fear that when people deny the infallibility or inerrance of Scripture and yet say it's "the Word of God", they are using "the Word of God" in a sense different from the one that historically and lexically attaches to the phrase. And that's misleading unless they are addressing a community that attaches that new sense to the phrase.

Two kinds of free will theodicies

I want to make a distinction that I've long thought is quite important, but which I think is not made enough. Suppose the theist responds to the problem of moral evil by saying that God's allowing moral evils is justified by the value of free will. There are two different ways that "the value of free will" could be invoked here.

First, the theodicy could depend on the intrinsic value of freedom itself. God allows me to freely choose the wrong thing, because my freely choosing something is intrinsically good qua free choice even when the thing I choose is bad.

Second, the theodicy could depend not on any value of the freedom in freely choosing the wrong thing, but only on the value that there would be in freely choosing the right thing were one to choose that. On this story, the only logically possible way God can have a chance of my freely choosing right over wrong in a given situation is by allowing me to have the possibility of choosing wrong over right in that situation. Notice that on this second story, if I end up freely choosing wrong over right, the evil of my choice might well be gratuitous in the sense that there is no compensating good for it. But even if the evil is gratuitous, God could be justified in permitting the evil to happen, since the only way he had to prevent it would have removed all possibility of my freely choosing right over wrong. This second story doesn't require there to be any intrinsic value in freedom as such. It only requires there to be value in freely choosing right over wrong.

I think the second story is superior to the first. But, interestingly, the second story is not available to a Molinist. For the second story only works when God cannot rely on knowing how you would choose in a situation when setting up the situation. (The story is available to a simple foreknowledge theorist if God cannot rely on knowledge of the future in ways that set up explanatory circularity.)

Another dice game for infinitely many people

Consider:

• Case 1: There are two countably infinite sets, A and B, of strangers. The people are all alike in morally relevant ways. I get to choose which set of people gets a lifetime supply of healthy and tasty food.

Clearly, it doesn't matter how I choose. And if someone offers me a cookie to choose set A, no harm in taking it and choosing set A, it seems.

Next:

• Case 2: Countably infinitely many strangers have each rolled a die, whose outcome I do not see. Set S6 is the set of people who rolled a six and set S12345 is the set of people who rolled something other than a six. The people are all alike in morally relevant ways. I get to choose which set of people gets a lifetime supply of healthy and tasty food.

Almost surely, S6 and S12345 are two countably infinite sets. So it seems like this is just like Case 1. It makes no difference. And if you offer me a cookie to choose S6 to be the winners, no harm done if I take it.

But now suppose I focus in on one particular person, say you. If I choose S6, you have a 1/6 chance of getting a significant good. If I choose S12345, you have a 5/6 chance. Clearly, just thinking about you alone, I should disregard any cookie offered to me and go for S12345. But the same goes when I focus on anybody else. So it seems that Case 2 differs from Case 1. If Case 1 is the whole story--i.e., if there is no backstory about how the two sets are chosen--then it really doesn't matter what I choose. But in Case 2, it does. The backstory matters, because when I focus in on one individual, I am choosing what that individual's chance of a good is.

But now finally:

• Case 3: Just like Case 2, except that you get to see who rolled what number, and hence you know which people are in which set.

In this case, I can't mentally focus in on one individual and figure out what is to be done. For if I focus in on someone who rolled six, I am inclined to choose S6 and if I focus in on someone who rolled non-six, I am inclined to choose S12345, and the numbers of people in both sets are equal. So I don't know what to do in this case?

Maybe, though, even in Case 3, I should go for S12345. For maybe instead of deciding on the basis of the particular situation, I should decide on the basis of the right rule. And a rule of favoring the non-six rollers in circumstances like this is better for everyone as a rule, because every individual will have had a better chance at the good outcome then?

Or maybe we just shouldn't worry about the case where you see all the dice, because that's an impossible case according to causal finitism? Interestingly, though Cases 1 and 2 only require an infinite future, something that's clearly possible.

Can a life of eternal pain be worth living?

Sally is in moderate pain and opts for a painless medical intervention that extends her life by one more day just because she wants to experience another day. Surely Sally is not being irrational. It's not irrational to choose to experience another day, even if that day involves moderate pain. Further, whatever the merits of a defense of euthanasia in the case of severe pain (in the end, I will reject euthanasia even in those cases), defending euthanasia in the case of moderate pain is implausible.

This suggests that it can be worth living for a finite amount of time in moderate pain. Moreover, it can be worth doing so even if there is no prospect of pain-free life afterwards. The rationality of Sally's decision doesn't depend on her beliefs about the afterlife. All this suggests a strong intuition that the experience of life, by itself, is enough to make life worth living despite moderate pain. But if it makes life worth living for a finite amount of time, why not an infinite?

Well, maybe an infinite life of moderate pain would result in extreme mental pain of hopelessness and ennui. But notice that this is a contingent consequence. A person who doesn't think much about the future can avoid the pain of hopelessness, and a person who doesn't remember having had many such days can avoid the pain of ennui. Thus it is logically possible to have a worthwhile infinitely long life of moderate pain without any great compensating goods besides the good of experiencing life itself, at least as long as one wasn't very thoughtful.

Thus, it is logically possible to have a painful but mildly worthwhile eternity in hell. It could be a life where enough of people's memories are wiped to prevent excruciating ennui, and where the people's minds have degenerated to a point where they don't care much about the future. (Would it be surprising if the minds of the damned weren't in tip-top shape?)

Now, one of the main reasons people reject the doctrine of hell is because they think that a loving and just God would not allow a person to exist for eternity in a state worse than nonexistence. But if it is possible to have a painful but mildly worthwhile eternity in hell, so that we need not suppose that eternity in hell is worse than nonexistence, this particular argument against hell disappears.

Objection: The biblical picture of hell involves not merely moderate but excruciating pain.

Response: Let's grant a literal picture of eternal burning. Now, being burned is normally an excruciating pain. Either divine goodness would rule out eternal excruciating pain or it wouldn't. If it wouldn't, the objection to hell disappears. But if divine goodness would rule out eternal excruciating pain, then it follows logically that if there is a God and eternal burning, then God does not allow that eternal burning to be eternally excruciating. Perhaps he provides fairly effective painkillers.

Eternal nagging, endless second chances and hell

Jabba the Hutt asks for a passionate kiss. You really don't want to do it and you don't. So he asks you again the next day. And again the day after. And so on. Each time Jabba asks you, there is some small chance you'll agree. Let's say that that chance is always at least one in a googolplex. Now suppose you and Jabba live forever. He asks you every day. Then by the Law of Large Numbers, it is nearly certain (i.e., has probability one or one minus an infinitesimal) that one day you will agree, no matter how disgusted you are by him.

The practical inevitability of the kiss means there is a sense in which your agreeing has been forced out of you by Jabba's eternal nagging, even though you were free on the particular occasion when you agreed to the kiss. We might say that you were quite free not to kiss on day n, where n is the day you actually kissed Jabba, but you were not really free never to kiss him. Yuck! How is it freedom when you are guaranteed to kiss a disgusting giant slug?

Now the two best alternatives to the traditional Christian doctrine that those who after a set deadline (death, say) opt against God are excluded from heavenly union with God are:

• Imposition: God imposes moral transformation on those who do not freely opt to love him.
• Endless Second Chances: God ensures that those who refuse him nonetheless always have another chance.

Here I take for granted Jerry Walls' argument that for a sinner moral transformation is metaphysically necessary for heavenly bliss, as heavenly bliss is constituted by a love relationship with God.

It's pretty plausible (pace compatibilists) that in Imposition, God takes away the agent's freedom to refuse him. But if the eternal nagging argument works, then in Endless Chances it looks like God all but takes away the agent's freedom, making it all but inevitable that the agent will eventually agree.

It is offensive to compare God to Jabba the Hutt. Yet for the person who is opposed to God, eternal union with God is subjectively rather like kissing Jabba the Hutt. Nor would it make the story more palatable if Jabba were to promise to make you enjoy the kiss, say by exuding pheromones or changing your preferences, as soon as you say "yes" to him. Of course, objectively God is infinitely lovable--but those have rejected him have set their hearts against that truth.

Objection: God could set up a version of Endless Second Chances on which it is not inevitable that the agent will agree by allowing each of the agent's refusals to affect the agent's character by even further lowering the chance of subsequent acceptance of God's offer. If the subsequent chances decrease sufficiently (say, by a half each time), the overall probability of eventually accepting might be significantly different from one.

Response: Yes, but this loses out on what I take to be one of the main merits of hell, that hell stops the agent's moral deterioration. On this picture, there is a significantly non-zero chance that the agent will continue morally deteriorating for eternity. And that's unfitting.

God is God's love

1. It is permissible to center one's life on God's love.
2. To center one’s life on anything other than God is idolatry.
3. Idolatry is impermissible.
4. So, God is identical with God's love.

A serious difficulty for some powers accounts of modality

On my causal powers account of modality, p is possible provided that either p is true, or something has an nth order power to make p true for some n. Here, a first order power to make p true is just a power to make p true. A second order power to make p true is a power to make there be a first order power to make p true. A third order power to make p true is a power to make there be a second order power to make p true. And so on.

Here's a serious problem. It seems quite possible that there is a sequence of false propositions p₁,p₂,p₃,... with the following properties:

1. It is possible that all the propositions are true.
2. For each n, something has an nth order power to make p_n true, but nothing has a lower order power to do so.

Now, let P be the proposition that all the p_n are true. Then P is false. But for every n it is true that nothing has an nth order power to make P true, as nothing has an nth order power to make p_n+1 true.

For concreteness, we might suppose that there are infinitely many planets, and on the nth planet there is a fertile asexually reproducing person x_n who has no children. Let p_n be the proposition that x_n has nth level descendants (where first level descendants are children, second level descendants are grandchildren, and so on).

What should I do? I can think of one option I don't like and two I can live with.

The option I don't like is to adopt strong assumptions about the nature of time that rule out the above story, such as an open future view plus discreteness assumptions.

The two I can live with both involve my scrapping the nth order power stuff. The first option is to make my thesis more modest: causal powers are the ground of metaphysical possibility, but I eschew giving an account of metaphysical modality in terms of causal powers. Then I can say that the possibility of P is grounded in powers, without giving a specific account. This is unattractive because it's unambitious.

The second option I can live with is to say that a thing can have a causal power to produce an effect even when it cannot directly produce the effect. Thus, I not only have a causal power to have children, but a causal power to have grandchildren. Then in the infinitely many planets scenario, the childless people jointly have a power to make P true. This seems to be the best option, but I really liked the iterated account.

Are elementary particles extended simples?

This argument is valid and every premise is plausible:

1. An elementary particle is located at every point where its wavefunction is non-zero.
2. An elementary particle is simple.
3. A simple located at every point of a region with non-zero volume is an extended simple.
4. Typical elementary particles have a wavefunction that is non-zero at every point of a region with non-zero volume.
5. So, typical elementary particles are extended simples.

Branchy gunk

An object is gunky provided that all of its parts have proper parts. Gunk is usually considered a really outré possibility. I want to offer some examples of intuitively conceivable gunky objects to broaden the philosophical imagination. The examples are all predicated on an Aristotelian ontology that allows for parts but denies other aspects of classical mereology. The thought behind the Aristotelian ontology of parts is that the parts of a thing correspond to natural functionally delineated subsystems. My heart is a part of me, as is my left arm. But there is no such part of me as "the left half of my heart" or "me minus my left arm".

Example 1: An infinite tree in 3D.

Here's a plausible Aristotelian thought about trees. Suppose that we have a branch A that splits into sub-branches B and C. Then branch A is an object that has both B and C as parts. However, there is no such part as A minus (B plus C). I.e., there is no object that consists of the part of A before the split. For the naturally delineated subsystem is the whole branch, including sub-branches, rather than the part of the branch without the sub-branches. Now imagine a fractal tree-like structure where the branches split into sub-branches, and the sub-branches into sub-sub-branches, and so on ad infinitum. Suppose, further, that there are no smaller natural functionally delineated subsystems than branches, sub-branches, sub-sub-sub-branches, etc. (This differs from real-world trees, which are made of cells.) The result is gunky: each part of the structure is a branch at some level, and each branch itself gives rise to sub-branches.

Dynamically, the structure can be thought of as built out of extended simples. We start with a trunk (a zero-level branch) that grows gradually. Then the trunk splits into branches. As a result, the trunk ceases to be simple: it has two or more proper simple parts, namely the branches, but it is not just the sum of the branches. The branches initially are simples, but eventually split themselves. If each step takes half the time of the preceding, after a finite amount of time we have the full infinite gunky tree.

Example 2: A four-dimensional example.

Suppose a spatial simple can survive becoming non-simple.(This was a governing assumption in the dynamical story in Example 1.) Suppose there are no proper temporal parts. Now, imagine we have a simple A, which survives becoming a non-simple made of two simples B and C. Then repeat the process with each simple. Continue ad infinitum, but don't require the process to speed up in any way. At any finite time, there are only finitely many objects. But the whole four-dimensional thing is gunky: A is made of B and C, B is made of D and E, and so on.

Example 3: Aristotelian temporal parts of a spatially simple thing.

On the Aristotelian ontology of parts, there won't be arbitrary temporal parts: there won't be the temporal part of me from my third to my fourth year. However, there might be naturally delineated temporal parts, like my adult part. Now imagine a person who never dies, and every five years receives a PhD in another discipline. If PhD-in-discipline-X counts as a naturally delineated temporal part, then the person will have a sequence of temporal parts like: doctor of biology, doctor of physics, doctor of chemistry, etc. Moreover, if we list these parts in the correct order, it gets gunk-like. If her first PhD is in biology and the second is in physics and the third is in chemistry, then the doctor of chemistry will be a part of the doctor of physics which will be a part of the doctor of biology. Moreover, there might be no such part as not-a-doctor-of-biology or not-a-doctor-of-physics (by the same token as on the Aristotelian story, there is no such part as me-minus-my-left-arm). Now, suppose that the person in question is an angel and hence has no spatial parts, and that the person has no significant temporal divisions besides the acquiring of PhDs. Then the individual is gunky: each part has a proper part. And this is easy to imagine, as long as we aren't worried about temporally extended simples.

Final remark: I don't know if these conceivable things are metaphysically possible.

If only God is perfect, then God has no proper parts

This argument is valid:

1. Only God is perfect.
2. Every part of God is perfect.
3. So, every part of God is identical with God.
4. So, God has no proper parts.

Premise (2) seems obviously true. So, we learn from the argument that if only God is perfect, then God has no proper parts.

Substances do not have substantial proper parts

It's an old maxim of Aristotelian metaphysics that substances do not have substantial proper parts. Here's an argument for it, in the case of material substances. Suppose a material substance A has a substance B as a proper part. Now, arguably A is wholly composed of two parts: the matter M and the form F.

Now the form G of B cannot overlap M, as then the form would be partly material. So G must be a part of F. But forms of substances are simples. So G must be all of F. But then we have two substances with the numerically same form, and that seems absurd.

A central assumption in the argument is that forms are simples. There may be a way of making an argument without that assumption. Suppose we say that G, the form of B, is a proper part of F, the form of A. Now if any proper part of a substance is a substance, then my heart is a substance--it's nicely delineated, and one of the best candidates. But I can survive the destruction of my heart (I would just need a machine to circulate the blood). And surely if my heart is destroyed, its form is destroyed as well. But my form doesn't seem to be intrinsically changed by the destruction of my heart. Yet if the form of the heart were a part of my form, then my form would be intrinsically changed by the destruction of the heart.

Does Christianity require a belief in matter?

The doctrines of incarnation, resurrection and real presence certainly require us to believe ordinary language existence claims about bodies, bread and wine. It's hard to take Scripture to be inspired without believing ordinary language existence claims about plants, animals, mountains, seas, etc. But do we need to believe that there is matter?

A search of the Church Councils up to and including the First Vatican Council turns up nothing dogmatic about "matter" in the relevant sense of the word (I am not including the technical sense of "matter of a sacrament" in sacramental theology). Searching for "material" finds some talk of material weapons, material flesh, and material food and drink. But I think that it would seem to me to be an overreach to take the Councils to be dogmatically teaching that weapons, flesh and food and drink are material. Rather, the relevant distinction seems to be between the spiritual weapons, spiritual flesh and spiritual food and drink and their ordinary earthly versions, rather than teach something about the nature of the ordinary versions, except that they differ from the spiritual.

I used to think that we need to believe hylomorphism. After all, the Fifth Lateran Council teaches that the soul is the form of the body. But while this gives us the morphê (form) part of hylomorphism, it doesn't give use the hyle (matter) part. We need to believe that the soul is the form of the body; not that it is the form of the matter.

If this reading of the Tradition is right, then Christian philosophers do not need to try to figure out the knotty question of what constitutes materiality. We have to accept, in some way, the existence of bodies, bread and wine, but we don't have to say that these things fall into some philosophically important kind like "matter". The handful of statements about "material" things we can simply understand in the vague way as about "things relevantly like ordinary things around us", without thinking that matter is any kind of metaphysically or physically important kind. We don't have to worry that if it turns out on our best science that physical reality is constituted by fields rather than particles, then we will have a conflict between faith and science. We still would have to find a way of locating bodies, bread and wine within physical reality, but we would not have to identify them with bits of matter.

Of course, it may turn out that the concept of matter has philosophical or scientific use apart from the needs of faith. But I have a suspicion that thinking about the nature of the body may be more promising than thinking about the nature of matter.

Two conceptions of matter

The philosophical tradition contains two conceptions of matter. One kind, associated with Descartes, connects matter with space: matter is what is responsible for spatial properties like extension or location. The other, associated with Aristotle, connects matter with passivity: matter is what makes an entity have a propensity to be the patient of causal influences. The spatial conception of matter has been the more popular one in recent times. But here is a reason not to go for the spatial conception of matter. The concept of materiality seems fairly close to the fundamental level. But it may well turn out--string theory is said to push in that direction--that at the fundamental level there is no such thing as space or time or spacetime. If that is a serious epistemic possibility, it would be good to do more work on the Aristotelian option.

General Relativity and the dimension of reality

When people first hear of General Relativity and the idea that spacetime is curved, they naturally imagine a higher-dimensional uncurved space in which our four-dimensional spacetime curvily sits. They may even be shown pictures of a two-dimensional sheet warping into three dimensions. But it's usually then explained to them that that's a misleading way to look at things: reality is just four-dimensional, but has a metric that makes it behave like it's curving in a higher-dimensional space.

What if the natural way of thinking about this is right? What if, say, reality is an 89-dimensional Euclidean space with signature (2,87), but physical objects are constrained to live on a 4-dimensional subset of it? The constraint could be effected, for instance, by a global discontinuous scalar field on the 89-dimensional space that takes two values: 1=allowed and 0=forbidden.

I suppose the main reason not to go for an ontology like this for General Relativity is that it's messier.

Are Leibniz's monads immaterial?

Leibniz says that "souls, like all other Unities of substances, are immaterial, indivisible and imperishable" (Leibniz's letter to Churfuerstin Sophie). These "Unities" are, of course, the monads as Leibniz explicitly notes earlier in the sentence. So Leibniz is claiming that monads are immaterial. I think Leibniz may be making a mistake in exposition of his own view here. It is essential to Leibniz's view that monads are spiritual. But there is a reasonable story to be told on which they are also material.

A plausible story is that to be material is just to have a place in space. But space on Leibniz's picture is just an abstraction from the interrelations of things in space. These interrelations are constituted by the harmoniously ordered interplay of the monads' representations of the universe. But these representations have--Leibniz is explicit about this--have a point of view. We can thus reasonably identify the location of a monad with the location of its point of view. Monads, then, have a place in space. If they have a place in space, then it seems we should say that they are material.

This was a bit too quick, though. First, it might be that some monads--God, for instance (though I don't know that Leibniz ever calls God a monad)--might have a point of view that is non-spatial in nature. Those monads won't be material.

Second, one might think that having a location is insufficient for spatiality. Two examples. First, God is a paradigm of an immaterial being, and yet the tradition holds that God is present everywhere. Second, on dualism, the soul is immaterial, and yet the soul might be said to be located wherever the body is.

The case of God is, I think, easily handled. Maybe materiality involves not just having location, but being locationally limited. Omnipresent beings aren't locationally limited. But those monads that have a single point of view that fits into the spatial order are locationally limited.

The case of the soul is, I think, a bit more difficult. One option is to say that the soul has its location derivatively from the location of something else--viz., the body. So our account of materiality now is: x is material provided that it has a limited location that does not derive from the location of something else.

Leibniz's monads qualify--or at least those that embody a spatially limited point of view. While the monads' location derives from their representations, it does not derive from the location of their representations--it derives from the interrelation of the representations. (Objection: The monad's location derives from the location of its point of view. Response: Leibniz's ontology does not include points of view as entities.)

Perhaps, though, there is something more to materiality than spatiality. Leibniz probably thought that extension is needed. Extension seems to be the occupation of multiple locations. In that case, Leibniz should have said that while individual monads are not material, in aggregate they are material. But I think requiring non-zero extension is a mistake. We might find out that all fundamental particles are unextended, and that shouldn't lead us to hold that they are immaterial.

Here's another move that Leibniz could take, though. He could say that if we try to spell out the definition of materiality, yes the monads do qualify. But it is unhelpful to put the ingredients of Leibniz's quite unique ontology into the straitjacket of other ontologies. Yet if for the sake of exposition we draw analogues, then we can say that Leibniz's monads are more like the immaterial elements of other ontologies than like the material ones.

Uniform motion and relationalism

An old objection to relationalism about space--an objection going back to the Leibniz-Clarke correspondence--is that it seems possible for all things to move together at the same speed in the same direction. But since the relations between things don't change when they all move together, on a relationalist view of space it seems impossible to make sense of global uniform motion.

Here's a solution to the objection: The motion of an object x can be characterized by saying that x at t₂ is at a non-zero spatial distance from x at t₁. This allows one to characterize absolute motion in a relationalist account of space, which has typically been held impossible.

The above story works most neatly if we have eternalism and temporal parts: then x moves provided that it has temporal parts at a spatial distance from each other. But we can also do this with eternalism without temporal parts, provided that we index distance relations to two times. Whether a presentist who is a relationalist about space can make use of the solution depends on how well the presentist can solve the problem of cross-time relations.

I don't personally like this story, because I would prefer a relationalism based on spacetime relations rather than spatial ones.

Murder and injustice

Not every kind of killing is a murder. Only an intentional killing is a murder. And not every intentional killing is a murder--just killing in a just war isn't murder. Only a wrongful intentional killing is a murder.

But not every kind of wrongful intentional killing is a murder. Jim takes a vow of non-violence, is drafted into the military notwithstanding the vow and intentionally kills in a just war in a way that is wrong due to violation of the vow (in some cases it might be that the needs of defending the innocent could override the vow, but stipulate that this isn't one of those cases). This intentional killing is a vow-breaking rather than a murder. Samantha is a police officer who shoots down a terrorist shooter who is on a rampage. However (and Samantha knew this), this terrorist is a crazy scientist who has just discovered a cure for a medical condition that kills millions, a cure that he was going to share after murdering ten people. Killing the scientist is wrong, but it's not a murder. Frederick is an executioner executing a duly convicted person who clearly deserves death, but this is a case the death penalty is impermissible for reasons other than justice (say, because there is a less violent way to protect society, which according to Evangelium Vitae implies that the death penalty is wrong). Martha intentionally kills an unjust aggressor in a war where her side meets some but not all the conditions of a just war: there is just cause, but the condition of reasonable expectation of success is not met.

In the above cases, the intentional killing is wrong but for reasons other than justice to the person killed. Indeed, in at least the cases of Jim and Samantha, the person being killed isn't being wronged at all.

I hypothesize that an intentional killing is a murder only if it is wrong as an injustice to the person killed. But this condition is still not sufficient. Suppose Jim instead of taking a vow of non-violence promised Patricia that he would never do anything to physically harm her, unless it was his moral duty to do so. And now Jim faces Patricia in a just war, under circumstances such that apart from the promise it would be permissible but not obligatory for him to kill her, and with the promise it is impermissible for him to kill her. Killing Patricia would be unjust to her, but the injustice is that of breaking a promise to her, rather than that of murder.

It's looking to me that murder is an intentional killing that is wrong due to a particular kind of injustice to the person being killed. It is difficult to specify the particular kind of injustice in a non-circular way, though.

Corollary: If suicide is a form of murder, then it is possible to be unjust to oneself.

Movement

The following seem quite plausible:

1. It is possible for an object both (a) to have both a first and a last moment of its existence and (b) to be moving at every time during its existence.
2. It is not possible for an object (a) to exist at only one time and yet (b) be moving.

By (2), movement is not an instantaneous property: it is not a property an object has solely in virtue of how it is at one moment. By (1), however, movement is not a property defined in terms of the past and present states of an object (say, "an object moves at a time provided that it is a different location from where it was in the past"), since it can move at the first moment of its existence; nor is it a property defined in terms of the present and future states of the object since it can move at the last moment of its existence.

So what is movement? We could say that an object is moving at time t provided that there are arbitrarily close moments t* at which the object is in a different location. This would make sense of both (1) and (2). But this account falsifies the following intuition:

3. If a ball is thrown vertically into the air, then at the high point of its flight it is not moving.

(If it were moving, would it be moving upward or downward?) For at moments arbitrarily close to that top-point time, the ball is at different locations.

We could try to define movement in terms of there being a well-defined non-zero derivative of the position with respect to time, with the derivative being one-sided at the beginning and end of the object's existence. But then, given continuous time (which we need anyway to have time-derivatives), an object could continuously change location without ever moving, since there are continuous nowhere differentiable functions.

So what should we say? I think it is that the concept of "moving at time t" is underspecified, and specifications of it simply aren't going to cut nature at the joints. Being at different places at different times (at least relative to a reference frame) makes good and fairly precise sense. But moving (or changing) at a time does not. Zeno was right about that much.

Marriage and the state

There is a presumption against the state imposing or enforcing restrictions on people's behavior. That's why, for instance, the state does not enforce private promises where money doesn't change hands. Now, marriage has two primary normative effects:

1. Make sexual union permissible;
2. Impose a rich tapestry of duties that the spouses owe to one another.

Most Western jurisdictions do not have a legal prohibition of fornication, however, which makes the first of the two primary normative effects moot with respect to the state (though of course marriage still is needed for sexual union to be morally permissible, as I argue in One Body). In those jurisdictions that do not legally prohibit fornication, the primary legal effect of marriage is entirely restrictive. Hence, in those jurisdictions, there is a presumption against the state's recognition of any marriages at all. (One might argue that the state needs to license marriages in order to render sex morally permissible; but marriage in the moral sense does not require state involvement.)

In those jurisdictions where fornication is not a crime, I think it is helpful to start debate about things like same-sex marriage or polygamy with a presumption against state involvement in any marriages whatsoever, and then ask in what cases, if any, that default negative judgment can be overcome.

(For the record, I do think the presumption can be overcome in opposite-sex cases, because of the connection with procreation. But I am not arguing for this here.)

Missing the center of the target with infinitely many arrows

Suppose that a countably infinite number of infinitely thin or perfectly symmetrical arrows is independently shot at a continuous target, with the distribution of impact points uniform over the target. (The independence requires that the arrows can go through each other--say, because they are made out of laser beams--or are removed between shots.) How probable is it that the exact center of the target will be hit by at least one arrow? In classical probability, the answer is zero. Intuitively, this is because the number of points on the target is a bigger infinity than the countably infinite number of arrows.

What if, instead, the number of arrows is greater than or equal to the number of points on the target? Unfortunately, the standard probabilistic model (a product space with the number of factors equal to the number of arrows) for the situation cannot answer that question: the probability of a point being hit by at least one of an uncountable infinity of arrows will be undefined. It would be interesting to see if there is any way of getting a non-arbitrary answer to the question outside of the standard model, say by putting some natural restrictions on which extensions of the model one allows.

Spacetime: Beyond substantivalism and relationalism

According to substantivalism, spacetime or its points or regions is a substance, and location is a relation between material things and spacetime or its points or regions. According to relationalism, location is constituted by relations between material things. Often, the two views are treated as an exhaustive division of the territory.

But they're not. Lately, I've found myself attracted to a tertium quid which I know is not original (it's a story other people, too, have come to by thinking about the analogy between location and physical qualities like charge or mass). On a simplified version of this view, being located is a determinable unary property. Locations are simply determinates of being located. This picture is natural for other physical qualities like charge. Having charge of 7 coulombs is not a matter of being related to some other substances--whether other charged substances or some kind of substantial "chargespace" or its points or regions. It's just a determinate of the determinable having charge.

This determinate-property view is more like the absolutism of substantivalism, but differs from substantivalism by not positing any "spacetime substance", or by making the locations into substances. Locations are determinates of a property, and hence are properties rather than substances. If nominalism is tenable for things like charge or mass, the theory won't even require realism about locations.

Minimizing the number of fundamental relations between concreta

An interesting metaphysics project that could use more work is to minimize the number of fundamental non-intentional relations between substances. How few can we do with? I think it would be really great if one could reduce the number of such relations to a handful of relations, or at least determinable families of relations. There is only one candidate really clear to me: causation. I think it would be an interesting research project to adopt the working hypothesis that causation is the one and only such relation and see how things go.

A lot of people will want to add parthood to this list, but I don't think a substance can be a part of another substance. (Parts are grounded in wholes, and substances are not grounded in other things.) Spatial relations like being seven meters apart (on relationalism about locations) and being located at (on substantivalism about locations) are a family of plausible candidates.

A new theory of limbo

A fairly standard libertarian response to the question about how people can freely choose right over wrong in heaven is this: They have a morally perfect character that makes them unable to choose wrong but this character is the result of choices in this life, choices that they could have avoided. Thus, the choice of right over wrong in heaven is derivatively free, with the freedom deriving from non-derivatively free choices in this life. For a nice development, see Timpe and Pawl.

I had a student ask the question how this works for those who as small children and who hence have not developed their character through free choices. Multiple answers are possible, but I wanted to offer one that yields a somewhat interesting theory of limbo. The theory of limbo holds that some people--those who die in infancy are often given as an example--have not had the kind of life of faith that is required for heaven but nonetheless have done nothing to deserve hell. They are, thus, in limbo: a happy state that, nonetheless, falls short of heaven.

Here, then, is a theory of limbo. Limbo is very much like heaven. In fact, those who are in limbo are a part of the same community as those in heaven, and there is no difference of location, but only of state. Those who are in limbo enjoy most of the joys of heaven: the beatific vision of God, union with wonderful people, flourishing human activity, etc. Their lives are very much like the lives of those who count as being in heaven. However, their choices of right over wrong are not free, because although they have the same morally perfect character that those in heaven do, in the case of those in limbo, that morally perfect character is not the result of their own free choices in this life--it is simply imposed on them by God. So they don't have the joy of knowing that these choices are free, and they don't have the joy of remembering how they freely formed their character, but otherwise they get to enjoy all the joys of heaven.

On this theory, it is better to be in a heavenly rather than limboic state, but the main joy of heaven--the beatific vision--is equally had by people in both states. The difference is solely that those in the limboic state lack the derivative freedom that those in the heavenly state have.

What I don't like about this theory is this: I have the intuition that God shouldn't force people to love him. But perhaps I should simply say: it is better to love freely, but loving unfreely is still good?

Rambling thoughts on probability and multiverses

Imagine a countably infinite multiverse where there are no spatial relations between the different universes, but it makes sense to talk of a common time sequence measured by the duration of time elapsed from the beginning of a universe. Suppose all the universes start out the same way: they each consist of a closed box containing an unstable particle with a one-year half-life that decays into a stable particle. The only change there is is the decay of the particle.

Every year "half" of these particles decay. But now notice an oddity: from the second year on, there is a sense in which there are no global change at all. No matter how many particles have decayed, the global picture will forever be this: a multiverse consisting of universes infinitely many of which consist of a box with an unstable particle and infinitely many consist of a box with a stable particle. In other words, there is no qualitative change in the multiverse. Interesting and somewhat paradoxical: it looks like reality is unchanging and yet there is change. We capture the change when we keep track of non-qualitative features: this box had an unstable particle, and now this same box has a stable particle.

Now let's add one person to each universe. The person doesn't get to look inside the box, but knows the general setup. Each year, the person's probability that her box contains an unstable particle goes down by a half.

Lesson 1: self-locating probabilities cannot be read off of the qualitative features of the present state of the universe--either history or numerical identity matters. (This is the same lesson that Ian has been pushing me to learn from an earlier post.)

Now, in the above story, the universe started out with all the particles in the unstable state. But what if instead the universe started out with infinitely many of the particles in the stable state and infinitely many in the unstable state? What can we say about each person's probability that her box contains an unstable particle? Each year after the first, P(unstable now | unstable a year ago) = 1/2. But what about the unconditional probability? How likely is it after, say, one year that the particle in one's box is unstable? Well, it's half of the probability that it was unstable to begin with.

But what is the probability that it was unstable to begin with? Well, if we had a bit of a backstory, we could answer that. Let's say that God creates each multiverse, and he flips a fair coin as he gets to each, and puts in a stable particle on heads and an unstable one on tails. In that case, the probability that the particle in the box was unstable to begin with was 1/2. But if God instead rolled a die, and put in a stable particle on six and an unstable one otherwise, the probability of initial instability is 5/6. In other words, the probability of initial instability depends on backstory. But what if there is no backstory at all? What if this multiverse came into existence ex nihilo for no cause at all? It is tempting to assign probability 1/2 to instability. But why that? Why not, instead, 1/6 or 5/6 or something else? The choice seems arbitrary and the honest thing to say, I think, is just that there is no meaningful probability.

Lesson 2: Probability of initial conditions requires some sort of structure. A chancy causal structure will do. Perhaps a spatial structure could do--if the boxes were in a single spaced, arranged on a line, alternating between stable and unstable, it might be reasonable to say that the probability that one has instability is 1/2. Might be, but I'm not sure.

But of course probabilities in subsequent years will depend on initial probabilities. If we cannot say anything about initial probabilities without more structure, we can't say anything about probabilities in subsequent years, except conditionally.

Does anything in Lesson 2 hang on us working in a multiverse? I doubt it.