Non-local real presence of Christ

Aquinas’s account of location for material substance is as follows. Ordinary material substances have a special accident—one more fundamental than all other matter-related accidents—he calls “dimensive quantity”, but which I will just call “dimensions”. This accident makes the object have a specific shape and size. The substance is then in a place provided that its dimensions are “commensurate with” that place—namely, provided that the dimensions are fitted into the place.

We can now say that a substance x is present in a place z in virtue of the following two presentness facts:

1. x is present in its dimensions D

2. the dimensions D are present in place z.

Furthermore, in the ordinary case, we can have an account of what “present” means in (a) and (b). A substance’s being present in dimensions D is just the substance’s having the dimensions D as an accident. And the dimensions being present in a place z is just their being commensurate with z. (This last one would bear more analysis, but that’s not my interest here.) When we have (a) and (b) with “present” grounded in this way—by having and commensuration respectively—we have what St Thomas call “local presence” or what we might call “ordinary physical location”.

Now, in the Eucharist the following happens according to Thomas. The substance of the bread turns into the body of Christ. The accidents of the bread miraculously remain in existence, but are no longer the accidents of any existing substance (whether bread or Christ). In particular, the dimensions D of the bread remain (and are a subject for the other accidents). And Christ’s body is present on the altar (say) because of the following two presentness facts:

3. Christ’s body is present in the dimensions D which are formerly of the bread

4. the dimensions D are present on the altar.

The ground of (d) just like that of an ordinary case of (b): the dimensions are commensurate with the place. But since Aquinas insists that Christ does not take on the accidents of the bread, the ground of (c) must be different from the ground of the ordinary case of (a): Christ does not have D. (In particular, Christ is not round and thin when transsubstantiation happens in a western Catholic Church.) Instead, Thomas says about (c) that Christ is “substantially” in the “foreign” dimensions D, but is not a subject of these dimensions, i.e., does not have them. As a result, we have the same structure of presence as in the ordinary case—it is mediated by dimensions—but because the grounds of the substance’s presence in the dimensions are different, this is not ordinary physical location any more.

In my 2008 paper, I gave up on figuring out what is meant by “substantial presence”, and indeed suggested that the problem is insoluble, and we should go for a different solution, one on which Christ has ordinary physical location on the altar. That solution may well be right, but I want to try out a more Thomistic solution—though the full story does not fit with everything Thomas says.

Consider the relationship between Seabiscuit and his accident of swiftness. This relationship has two features which make for interdependence:

1. Seabiscuit’s swiftness ontologically depends on Seabiscuit, i.e., Seabiscuit ontologically sustains his swiftness

2. Seabiscuit is qualified by his swiftness.

In all ordinary cases, the relations of ontologically sustaining and being qualified by between a substance and an accident are coextensive. Seabiscuit sustains all his accidents and they all qualify him, and similarly for all substances. To have an accident is then for the accident to ontologically depend on one and for one to be qualified by it.

Expanded out this way, we have a richer story as to what grounds an ordinary substance being present in its dimensions: the substance ontologically sustains the dimensions and is qualified by them.

Now, Thomas’s denial that Christ is “subject to” the dimensions of the bread means one aspect of the ordinary relationship between a substance and its accident is absent here—Christ is not qualified by the dimensions. However, that still leaves the possibility that the other aspect of the relationship is present. In other words, we can suppose that miraculously Christ’s body ontologically sustains “foreign” accidents that this body is not qualified by. This ontological sustenance relationship makes Christ’s body be substantially in the accidents, including in the dimensions.

Thus, the expanded account is:

1. Christ’s body is present in the dimensions D formerly of the bread by ontologically sustaining these dimensions in the way that a substance sustains its accidents but without D qualifying Christ’s body and hence without D becoming its accident.

2. The dimensions D are commensurate with a place, just as in ordinary physical location.

The presence in (1) is a special case of a type of presence that Thomas recognizes in his account of divine omnipresence. Thomas says that God is present to all things “by his essence”, namely by directly being their cause. This causation is, of course, divine sustenance. Thus, Christ’s body’s being “substantially” in the dimensions by sustaining them ontologically is like God’s being “by essence” present to all things by sustaining them. This is a recognized and metaphysically serious mode of presence, and hence it plausibly counts as a real presence.

At this point, it may seem that I have solved the problem of what Aquinas means by the substantial presence of Christ’s body in the dimensions of (the former) bread. But there is one hitch. I think Aquinas disagrees with my account. When Aquinas discusses how the accidents of bread and wine can remain without their substances, his answer is not that the body of Christ sustains them, but that God sustains them, because anything that can be done by creaturely causes can be done by God. St Thomas’s phrasing very much sounds like he thinks the sustenance of the accidents is done directly by the power of God.

The account I am offering requires that God miraculously bestow on Christ’s body the power to sustain accidents foreign to it (without being qualified by them). I don’t see any good reason to think this can’t happen. We thus have an extension of Thomas’s account, but it is one that I think is compatible with other aspects of his metaphysics and theology.

I am still not completely convinced that I should abandon my account on which Christ’s body is present in the Eucharist by ordinary physical location. My account clearly makes Christ’s body by really present. The modified Thomistic account may do that, but it may not.

I want to end with a consideration in favor of the Thomistic view that Christ’s presence in the Eucharist is not ordinary physical location. Many Protestants think that Christ’s body is “spiritually present”, and historically the Reformed wing of the Protestant tradition has taken spiritual presence quite seriously—not just as symbol—while denying ordinary physical presence (I am grateful to one of my graduate students for pointing this out). An account of Christ’s real presence that makes Christ not be ordinary physically present thus has an advantage: it fits with the intuitions not only of many Catholic thinkers but also of many non-Catholic ones. Perhaps the modified Thomistic account just is what spiritual presence is, and hence we have a way of moving Catholics and some Protestants closer together through St Thomas.

Transsubstantiation and the conversion of bread into Christ's body

One of the philosophical challenges of Aquinas’ account of transsubstantiation is his insistence that the bread and wine are not merely annihilated and replaced by Christ’s body and blood, but that they are changed into Christ’s body and blood.

Now, it is easy to see how bread could be changed into a part of Christ’s body. That routinely happened when Christ ate bread in his earthly life. But Aquinas thinks that Christ is wholly present in the Eucharist, so that can’t be the account. But it is very puzzling what it would mean for an item B to be changed into an individual item C that already existed prior to the change. What would it mean, for instance, for the chair I am sitting on to change into the laptop I am typing this on? It is easy to imagine God moving the fundamental particles of the chair into positions such that they constitute a laptop. But that would be a case of the chair changing into a second laptop, not into the laptop that I am typing this on. Indeed, it seems like it’s impossible for something to change into something that already exists, simply because the thing already exists.

Aquinas is well aware of this objection, and has a fascinating response:

A form cannot be changed into another form, or one [designated] matter into another [designated] matter, by the power of a finite agent. However, such a conversion can be effected by the power of an infinite agent, which has an action on the whole entity. For the common nature of being belongs to each form and to each [designated] matter, and the author of being (auctor entis) is able to convert what there is of being in the one (id quod entitatis est una) into what there is of being in the other (id quod est entitatis in altera), by removing that by which it was distinguished from the latter.

Here is what I think is going on. Like many other philosophers before and after him, Aquinas thinks that individual objects need something whereby they are individuated—something that distinguishes them from other things. The project of figuring out what individuates things from other things is indeed a major part of Aristotelian metaphysics. Aquinas’ point seems to be this. God wields a very fine scalpel at the level of being, a scalpel so infinitely sharp that no finite being can wield it. That scalpel allows God to slice off an individual B that which distinguishes B from an individual C. When God slices that off, B literally loses its identity, and becomes C, as there is then nothing whereby B can be distinguished from C. (That God has such a fine scalpel is also indicated by the way that in the Eucharist he can slice a substance away from its accidents, and have the accidents remain without the substance.)

Let’s explore this account. First note that it seems to commit Aquinas to a different account of the individuation of material objects from his usual one. Aristotelians normally think that material objects are distinguished either by having different forms or, when the form is the same, by having different matter. Now, the bread on the altar and the body of Christ do have different forms: one is bread and the other is a human body. So on the usual Aristotelian account of what makes the bread different from the body of Christ, it is the bread’s bready form. But slicing away the bready form does not turn the bread into the body of Christ, or indeed into any human body. It just turns the bread into a formless lump of matter.

Perhaps we should suppose, however, that there is more than just literal removal going on. Maybe what happens is that God removes the bready form and replaces it with the form of the human body. But great as that miracle would be, that would just turn bread into a human, not into this human, Jesus Christ.

What if we suppose that God removes the bready form and replaces it with the form of Jesus Christ (namely, the soul of Jesus Christ)? But now the bread is simply becoming a new part of the body of Christ (in a miraculous verison of the way that the bread you may have for lunch may become new cells in you), and so only a part of Christ is present in the Eucharist.

But perhaps what I have described doesn’t slice away enough. Suppose the following happens. God slices the bready form away from the bread. That still leaves the bread’s matter. And the matter of the bread is distinct from the matter of Christ’s body. God continues removing the grounds of distinctness. He wields his infinitely sharp scalpel and carefully removes that in the matter of the bread which makes it be distinct from the matter of Christ’s body. The result is that that now the matter of the bread is not distinct from the matter of Christ’s body. Indeed, the matter of the bread literally converts into the matter of Christ’s body, not merely a new part of Christ’s body.

A major problem with this interpretation is that the form of bread is annihilated, whereas Thomas thinks the form of bread is also converted into the form of Christ’s body (admittedly with a qualification; see ST III.75.A6repl2 for details).

But perhaps we should make another move. Suppose that we have a non-Aristotelian account of individuation that works as follows: for any two created things, B and C, there is a relation that B has to C that individuates B from C and a relation that C has to B that individuates C from B. We can imagine each created thing having a vast number of labels. Somehow Alice has written into her being “I am not Bob” and “I am not Seabiscuit” and “I am not Oak Tree #18289”, and Bob has written into his being “I am not Alice” and “I am not Seabiscuit” and “I am not Oak Tree #18289”. This relational account of individuation does not require form or matter. It is not very Aristotelian. But it has a great theological merit: it makes the individuation of creatures be an image of the individuation of persons in the Trinity, which also proceeds (according to Western Christians) by opposed relations. Now imagine that God slices off of Alice the label “I am not Seabiscuit.” Instantly, Alice is converted into Seabiscuit. (Of course, it’s not right to say that Alice is Seabiscuit now. In this respect, it’s like when Bucephalus turned into a cadaver: Bucephalus and the horse-shaped cadaver are distinct entities.)

The exegetical problem with this interpretation is that it forces one to reject the standard Aristotelian story about individuation across species being by form and within a species being by matter. Instead, individuation is always by “individuating relations”. I am happy with this, because I never liked the standard Aristotelian story. But it makes it unlikely that the story is what Thomas has in mind.

But suppose one wants this to be more Aristotelian. Here is a way to do this. Take the orthodox Aristotelian account that across species individuation is by form and within species by matter. This account leaves unanswered the question of what makes a human form and a horse form different, as well as the question of what makes Peter’s matter different from Paul’s matter. Suppose we answer these questions by the relational account, thereby combining the Aristotelian account with the relational. Thus, a human form has (perhaps primitive) distinctness relations to all other kinds of forms, and Peter’s matter has a (perhaps primitive) distinctness relation to all other chunks of matter.

We can now imagine the following happening. There is bread on the altar. At the moment of consecration, God (a) removes from the bready form that which distinguishes it from a human form and (b) removes from the bread’s matter that which distinguishes it from Christ’s matter. Step (a) ensures that now the bread has human form, while step (b) ensures that the human form is that of Christ, since within a species the numerical distinction of forms is due to matter.

This last account is quite Aristotelian, and only requires that we go one step further than Aristotle by supposing an answer to the question of what makes different kinds of forms different and distinct chunks of matter distinct. It’s too Aristotelian for my taste—I don’t want matter to play that much of a metaphysical role. But it is a cool account, I think. And it could be Aquinas’.

The Real Presence and Relativity Theory

Jesus Christ like all human beings has an internal clock. One can measure that clock in heartbeats or in lower level physical interactions or in some other way. Let’s measure it in “internal years”. If Jesus was born in 4 BC, then in 4 BC, his internal clock was at about a year (he was conceived about 0.75 years before he was born). In 1 BC, it was at 4 years, in 1 AD, it was at 5 years, and in 30 AD, it was at 34 years.

I don’t know what Jesus’s internal clock was at in 100 AD, and it’s not immediately obvious that the question makes sense. For it is not immediately obvious that there is a correlation between time on earth and time in heaven of such a sort it makes sense to ask “What is happening in heaven right now?” After all, according to Relativity Theory, it doesn’t make sense to ask “What is happening right now in the Andromeda Galaxy” without specifying a reference frame for the “right now”, and it’s not immediately clear that there is a common reference frame between heaven and earth.

However, the real presence of Jesus in the Eucharist does provide a temporal correlation between heaven and earth. Around 22:45 UTC today, Jesus will come to be present in our campus parish. Moreover, Jesus will be present as an adult glorified human, not as the three-year-old he was in 1 BC. There thus appears to be a fact of the matter as to what his internal clock will be showing when he comes to be present at 22:45 UTC in Waco today.

Interestingly, this gives a temporal ordering on events scattered across the earth apparently independent of our ordinary relativistic reference frames. For if the Eucharist is celebrated around 22:45 UTC in Waco and around 22:45 UTC in London, there is a fact of the matter whether Christ as present in Waco is older or younger or at the same age (according to his internal clock), and this fact provides a reference-frame independent temporal ordering between these two Eucharistic celebrations.

Indeed, since according to Catholic and Orthodox faith, Christ remains Eucharistically present in the tabernacles across the world, we constantly have a temporal ordering between events scattered spatially across the world. In principle, this defines a theologically privileged reference frame between scattered events—a Eucharistic reference frame. Events at locations z₁ and z₂ in spacetime are Eucharistically simultaneous, we might say, provided that Christ as Eucharistically present at z₁ and at z₂ has the same value of the internal clock.

Of course, some philosophers of time think there is an objective reference frame in the physical world. If they are right, then very likely the theologically privileged frame is the same as the objective one.

All that said, it is not completely clear to me that Christ as Eucharistically present has to have a well-defined value of his internal clock. But I suspect so, because of the intuition that it is the adult and not toddler Jesus who is Eucharistically present.

Eucharist talk and other YouTube things

I just noticed that a talk I gave on the Eucharist is available on YouTube. And there is a convenient playlist of a number of talks I've given.

Transsubstantiation and magnets

On Thomistic accounts of transsubstantiation, the accidents of bread and wine continue to exist even when the substance no longer does (having been turned into the substance of Christ’s body and blood). This seems problematic.

Here is an analogy that occurred to me. Consider a magnet. It’s not crazy to think of the magnet’s magnetic field as an accident of the magnet. But the magnetic field extends spatially beyond the magnet. Thus, it exists in places where the magnet does not.

Now, according to four-dimensionalism, time is rather like space. If so, then an accident existing when its substance does not is rather like an accident existing where its substance does not. Hence to the four-dimensionalist, the magnet analogy should be quite helpful.

Actually, if we throw relativity into the mix, then we can get an even closer analogy, assuming still that a magnet’s field is an accident of the magnet. Imagine that the magnet is annihilated. The magnetic field disappears, but gradually, starting near the magnet, because all effects propagate at most at the speed of light. Thus, even when the magnet is destroyed, for a short period its magnetic field still exists.

That said, I don’t know if the magnet’s field is an accident of it. (Rob Koons in conversation suggested it might be.) But it’s comprehensible to think of it as such, and hence the analogy makes Thomistic transsubtantiaton comprehensible, I think.

A Thomistic argument for the possibility of an afterlife for animals

1. Accidents are more intimately dependent on substance than substantial forms on matter.

2. If (1) is true and God can make accidents survive without the substance, then God can make forms survive without matter.

3. If God can make forms survive without matter, then God can ensure life after death for animals by making their forms survive and restoring their matter.

4. God can make accidents survive without the substance.

5. So, God can ensure life after death for animals.

The most controversial claim here is (4), but that follows from the Thomistic account of the transsubstantiation.

Of course, there is a great gap between the possibility of an afterlife for an animal and its actuality. And the above argument works just as well for plants and fungi.

A horizontal aspect to transsubstantiation

The Eucharist has the vertical dimension of our union with Christ and a horizontal dimension of our union with our fellow Christians. The doctrine of transsubstantiation ensures the vertical dimension in an obvious way. But yesterday, while at a Thomistic Institute retreat on the Eucharist, I was struck by the way that transsubstantiation also deeply enhances the horizontal dimension of the Eucharist as a common meal.

Normally, in a common meal we eat together. Sometimes we eat portions cut from one loaf or carved from one animal, and that makes the meal even more unifying. But according to transsubstantiation, in the Eucharist we have a common meal where miraculously we each eat not just a portion of the same food, but the numerically very same portion: the whole of Christ. That is as deep a unity as we can have in eating.

Consider how there is less unity on the main alternatives to transsubstantiation:

• On symbolic views, we eat and drink different portions of bread and wine with the same symbolism.

• On consubstantiation, we eat the same Christ along with different portions of bread and wine.

• On Leibnizian views (where the bread and wine becomes a part of Christ), we eat different parts of the same Christ.

The transsubstantiation view has as much substantial unity in what is eaten as is logically possible. (Though there is some accidental disunity, in that the accidents—shape, color, position—are different for different communicants.)

Eucharist talk

I gave a Thomistic Institute talk on the Real Presence today. Here are the slides and here is the audio. I am particularly pleased with my argument against the purely symbolic view of the Eucharist.

Leibniz's idealistic transsubstantiation

I’ve been thinking how much nicer Leibnizian idealism is than the Berkeleyan sort, because you get this nice dose of realism from unconscious perception.

For instance, in one of his letters to Des Bosses, Leibniz offers a neat idealistic account of transsubstantiation: the unconscious perception of the micro-structure of the bread and wine perishes and is replaced with the unconscious perception of the micro-structure of Christ’s body, while the conscious perception of the macro-structure of the bread and wine remains. Material substance is better identified with micro-structure than macro-structure, and the macro-structure is more accident-like, so this counts as a replacement of the material substances in the bread and wine with the material substance of Christ’s body and blood.

Clever! But I am not sure what Leibniz can do with issues about size. The phenomenal perception of the micro-structure of Christ’s body presumably covers a larger volume of perceptual space than the macro-structure of the host. But Christ’s body is supposed to be where the host is.

Leibniz doesn’t say that this account of transsubstantiation is good. He suggests it’s the best one that can be adopted by Jesuits who don’t believe in composite substances.

Materiality and spatiality

I’ve been fond of the theory that materiality is just the occupation of space. But here is a problem for that view.

I have argued previously that we should distinguish between the internal space (or geometry) of an object and external space. Here is quartet of considerations:

• Imagine a snake one light-year in length out in empty space arranged in a square. Then imagine that God creates a star in the middle of the square. The star instantly disturbs the geometry of space and makes the distances between parts on opposite sides of the square be different from what they previously where. But this does not make any intrinsic change to the snake until physical influence can reach the snake from the star, which will take about 1/8 of a year (the sides of the square will be 1/4 light-years, so the closest any part of the snake is to the center is 1/8 light years). The internal geometry of the snake differs from the external one.

• We have no difficulty imagining a magical house whose inside is larger than its outside.

• Christ in the Eucharist has very different (larger!) internal size and geometry from the external size and geometry of where he is Eucharistically located.

• Thought experiments about time travel and the twin paradox suggest that we should distinguish internal time from external time. But space is like time.

Now, if internal and external space can come apart so much, then it is plausible that an object could have internal space or geometry in the absence of any connection to external space. Furthermore, if a material object ceased to have an occupation relation to external space but retained its internal geometry, it would surely still be material. Only a material object can be a cube. But a cubical object could remain a cube in internal geometry even after losing all relation to external space. But if so, then materiality is not the occupation of external space.

In fact, even independently of the above considerations about internal and external space, it just doesn’t seem that objects are material in virtue of a relation to something beyond them—like external space.

So, it seems, objects aren’t material in virtue of the occupation of external space. Could they be material in virtue of the occupation of internal space? Not substances! A substance does not occupy its internal space. It has that internal space, and is qualified by it, but it seems wrong to say that it is in it in the sense of occupation. (Perhaps the proper parts of material substances do occupy the substance’s internal space.) But some substances, say pigs or electrons, are material. So materiality isn’t a function of the occupation of internal space, either. And unless we find some third sort of space, we can’t say that materiality is a function of the occupation of space.

Perhaps, though, we can say this. Materiality is the possession or occupation of space. Then material substances are material by possessing internal space, and the proper parts of material substances are material by occupying the substance’s internal space. On this view, the materiality of me and my heart are analogically related—a fine Aristotelian idea.

But I have a worry. Point particles may not exist, but they seem conceivable. And they would be material. But a point particle doesn’t seem to have an internal space or geometry. I am not sure what to say. Perhaps, a point particle can be said to be material by occupying external space (in my proposed account of materiality, I didn’t specify that the space was internal). If so, then a point particle, unlike a square snake, would cease to be material if it came to be unrelated to external space. Or maybe a point particle does have an internal zero-dimensional space. It is hard to see what the spatiality of this “space” would consist in, but then we don’t have a good account of the spatiality of space anyway. (Maybe the spatiality of an internal space consists in a potentiality to be aligned with external space?) And, finally, maybe point particles that are points both externally and internally (particles that have non-trivial internal geometry but that are externally point-like aren’t a problem for the view) either aren’t material or aren’t possible.

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.

A Catholic argument against presentism

1. If Catholic teaching on the Eucharist is true, then there is a time t such that an accident of bread exists-at-t and no substance of which it is an accident exists-at-t.
2. It is metaphysically impossible that an accident exists and no substance of which it is an accident exists.
3. If presentism is true and x exists-at-t, then it is true at t that x exists.
4. Catholic teaching on the Eucharist is true.
5. If presentism is true and Catholic teaching on the Eucharist is true, then it is true at some time t that an accident of bread exists and no substance of which it is an accident exists. (By 1 and 3)
6. There is no time t such at that time it is true that an accident of bread exists and no substance of which it is an accident exists. (By 2)
7. So at least one of presentism and Catholic teaching on the Eucharist is false. (By 5 and 6)
8. So presentism is false. (By 4 and 7)

Traditionally, Catholic philosophers have denied (2). But (2) is very plausible, so we have a good argument against presentism.

Colors and transsubstantiation

This is going to be very speculative, and I doubt it yields an orthodox account of transsubstantiation, but since there is some chance that it does yield such an account (and if it doesn't, we might get a deeper picture of transsubstantiation by thinking about why it fails), it's worth thinking about.

Let's say, as a first approximation, that an object is white at a spacetime region U provided that the object has a direct causal power of reflecting light incident on U diffusely and approximately uniformly across the visible spectrum. Observe that in this definition nothing was said about U being a region that is occupied by the object. It is logically possible for an object to have a causal power of action at a spatial and/or temporal distance, thereby diffusely and approximately uniformly reflecting light incident on a region unoccupied by the object. Now suppose that a white piece of bread is going to be destroyed, but just before it is destroyed the causal power of whiteness that it has is enhanced to work at a temporal distance, thereby diffusely and approximately uniformly reflecting light incident on a spatial region shaped like a piece of bread in the future after the destruction of the piece of bread. Then there is a sense in which the whiteness of the piece of bread persists after the destruction of the piece of bread.

It seems there are two senses in which we can say that the whiteness of an ordinary object is at a location V. One sense is that the relevant causal power is located at V and the other sense is that the object is directly causing light to be reflected whitely at V. The location of the accident of whiteness can be identified either with the location of the causal ground of the reflection or with the location of the immediate effect of that causal ground (the second matches how Aquinas understands the locations of angels: they are deemed present where they act). Normally, the two locations coincide or are very close together. So there is a a sense in which, in the scenario where the bread has the power of causing white reflections after its destruction, the accident of whiteness exists at the location where the reflection occurs, and hence continues to exist after the destruction of the bread.

Accidents outliving their substances

Thomas Aquinas's take on transsubstantiation supposes that the accidents of bread and wine can continue existing even after the bread and wine have perished, something that was heavily criticized by people like Jan Hus.

But here is an argument for the possibility of an accident outliving its substance. Consider a very long rattlesnake, stretching out to maybe ten million kilometers in length. The rattlesnake is rattling for one second. The rattling of the tail is an accident of the rattlesnake, call this accident R. Then the snake is near-instantaneously destroyed, e.g., by a series of synchronized explosive charges.

Well, near-instantaneously in one reference frame! This snake is long enough that there will be another reference frame in which the front half is destroyed 15 seconds before the back half is. In this reference frame, there will be a time when the rattling of the tail occurs even though the front half of the snake doesn't exist. But a snake whose front half has been destroyed is no longer existing. So in this reference frame the accident R exists even though the snake no longer does.

Granted, in the case of the snake it is only true in some reference frames that the snake doesn't exist while R does, while in the Eucharist the persistence of the accidents past the demise of the bread and wine takes place in all reference frames. But once we have seen that the principle that accidents must be contemporaneous with their substance is not generally true, I think some wind is taken out of the objector's sails.

What is location?

I will first argue that location is a multiply-realizable—i.e., functional—determinable. Then I will offer a sketch of what defines it.

A multiply-realizable determinable is one such that attributions of its determinates are grounded in different ways in different situations. For instance, running a computer program is multiply realizable: that something is running some algorithm A could be at least partly made true by electrical facts about doped silicon, or by mechanical facts about gears, or by electrochemical facts about neurons. Moreover, computer programs can run in worlds with very different laws from ours.

As a result, a multiply-realizable determinable is not fundamental. But location seems fundamental, so what I am arguing for seems to be a non-starter. Bear with me.

Consider a quantum system with a single particle z. What does it mean to say that z is located in region A at time t?[note 1] It seems that the quantum answer is: The wavefunction (in position space) ψ(x,t) is zero for almost all x outside A. And more generally, quantum mechanics gives us a notion of partial location: x is in A to degree p provided that p=∫_A|ψ(x,t)|²dx, assuming ψ is normalized. On these answers, being located in A is not fundamental: it is grounded in facts about the wavefunction.

But it is also plausible that objects that do not have wavefunction can have location. For instance, there may be a world governed by classical Newtonian mechanics, and objects in that world have locations but no wavefunctions. (And even in a world with the same laws as ours, it is possible that some non-quantum entity, like an angel, might have a location, alongside the quantum entities.) Thus, location is multiply-realizable.

Very well. But what is the functional characterization of location? What makes a determinable be a location determinable? A quantum particle is located in A provided that ψ vanishes outside A. But a quantum particle also has a momentum-space wavefunction, and we do not want to say that it is located in A provided that the momentum-space wavefunction vanishes outside A? Why is the "position-space" wavefunction the right one for defining location? Why in a classical world is it the "position" vector that defines location, rather than, say, the momentum vector or an axis of spin or even the electric charge (a one-dimensional position)?

I want to suggest a simple answer. Two objects can have very similar electric charges, very similar spins or very similar momenta, and yet hardly be capable of interacting because they are too far apart. In our world, distance affects the ability of objects to interact with one another. Suppose we say that this is the fundamental function of distance. Then we can say that a determinable L is a location-determinable to the extent that L is natural and the capability of objects to interact with one another tends to be correlated with the closeness of values of L. This requires that L have values where one can talk about closeness, e.g., values lying in a metric space. In a quantum world without too much entanglement and with forces like those in our world, the wavefunction story gives such a determinable. In a classical world, the position gives such a determinable.

(One could also have an obvious relationalist variant, where we try to define the notion of being spatially related instead. The same points should go through.)

Notice that on this story, it may be vague whether in a world some determinable is location. That seems right.

I think this story fits well with common-sense thought about distance and location, and helps explain why we maintained these concepts across radical changes in physical theory.

This story also reminds me a little of what Aquinas says about the location of angels: An angel is at location x if and only if the angel is causally interacting with something at x. But there is a difference. While Aquinas defines location of an angel in terms of actual interaction, I define location at two removes from interaction: I only talk of the capability for interaction and I do not define location in terms of that, but in terms of a fairly natural determinable closeness in respect of which tends to correlate with capability for interaction.

We have two non-philosophical tests for a theory of location. One is whether it coheres with science and the other is whether it coheres with theology, and especially with transsubstantiation. It is no surprise that this theory coheres with science, since it was designed to. What about transsubstantiation (John Heil referred to it as a supercollider for metaphysical theories in medieval times)?

I think it coheres with transsubstantiation quite well. The Catholic tradition tends not to talk about physical but sacramental presence. This is a real presence of course. There are multiple ways of being located for Aquinas: physically, by power (as angels are where they act and as God is everywhere), as well as this sacramental presence that he has an elaborate metaphysical theory of (see my paper here). Our above theory allows for a multiplicity of location determinables, all taking values in a common space, and so there might be some determinable that would give an account of sacramental presence.

But one could also defend the physical presence of Christ's body in the Eucharist coherently with my view. Christ's body could have multiple wavefunctions, each defining a different location. Or maybe space could curve in on itself as I suggest (but do not endorse as my preferred view) in the above-cited paper so that the location of the consecrated host and the location of heaven are literally the same.

So while I didn't design the view initially with the transsubstantiationon in mind, I think it passes the transsubstantiation test.

"Wholly present"

Here's something I've been thinking about. I want to start with the technical notion of being located at a region. This notion allows for partial location. If I have one leg in Arkansas and one in Texas, then I count as located in Arkansas and located in Texas. If regions have points, then I am located at a region if and only if I located at some point in the region (maybe that's a more primitive notion).

I'd like to move from the notion of being located at a region to the notion of being wholly present in a region. I am now wholly located in Texas, but if I had a leg in Arkansas and a leg in Texas, I would be wholly located in neither.

I could take the notion of being wholly present as primitive. But I don't want to do because it's a three-dimensional notion, while I think I am a four-dimensional entity, so to me it's a derivative notion.

One obvious thing to say is:

1. x is wholly present in A iff x is located at A and not located outside A.

This rules out multilocation—being wholly present at two distinct regions—by fiat. In so doing, it rules out both my view of the doctrine of transsubstantiation (since on my view, it is literally true that Christ is wholly spatially present in different places[note 1]) and the possibility of backwards time travel (since if you can travel back in time, you could be wholly present in two places, and shake hands with yourself).

A plausible move is to introduce parts or, more generally, features (the blueness of my eyes is a feature but not a part) and their locations (I stipulate that every part, proper or not, is a feature). Maybe, then:

2. x is wholly present in A iff every feature of x is located at A.

While this works for transsubstantiation, it doesn't work for time travel. For suppose that tomorrow I lose a leg, and I travel back to today, so that I am in another room in addition to this one. Then it is false that every part of me is located in that other room, since the lost leg isn't there.

There is another interesting problem with (2) and time travel. Suppose that in ten seconds I travel back to the present, so that I am wholly present in two disconnected rooms, and suppose that in the ten seconds I have neither lost nor gained any features. Let A_L be the left half of the space occupied by me in one of the rooms and B_R be the right half (including the cut line—don't put the cut line in A_L) of the space occupied by me in the other room. Let C be the disconnected region that is the union of A_L and B_R. Then by (2) I am wholly present at C as every one of my features is in either A_L or in B_R or in both. But surely I am not wholly located in the messy region C.

At this point things get difficult. My best solution today is moderately complex (but not as complex as my best solution yesterday). It requires the introduction of two sets of times for a persisting substance. There are internal times, which correspond to the internal development of the substance, and there are external times, which correspond to what goes on in the external world. Normally, the two are nicely correlated. But time travel discombobulates things. If in one minute I travel 24 hours into the future, then in one internal minute I will be 24 external hours forward. And if in a minute I travel 24 hours into the past, then in one internal minute it will be externally yesterday.

Now, take the case where I am right now in room A in the normal way, but in room B due to having time-traveled back to that room after losing a leg. Let T be the present external time. There are two internal times associated with t. At internal time t₁, I am in room A, and at internal time t₂, I am in room B. Moreover, at t₁, I have two legs, though at t₂ I only have one. I guess at the external time T, I have two legs. My being wholly present in B does not require that I have both of my legs in B. It only requires that I have in B all the legs that I have at the internal time t₂.

This yields the following pair of definitions:

3. x is wholly present in A at its internal time t iff every feature that x has at t is located at A at t.
4. x is wholly present in A at external time T iff there is an internal time t of x such that (a) x's internal time t is externally at T and (b) x is wholly present in A at t.

This gives the right answers with respect to (a) transsubstantiation, (b) time-travel and loss/gain of parts, and (c) time-travel and the union of the A_L and B_R regions.

The account does, however, have the consequence that if x is an extended simple with all features spread over all of x (so, x is the same color all over, etc.), then it counts as wholly present at every point at which it is located. This consequence is perhaps not so plausible, but I can live with it.

The deep question for the philosophy of spacetime

There is more than one way of putting this point, so the assumptions I will make are not at all essential, and I don't even endorse the assumptions. Assume absolutism about spacetime. On one reading of absolutism, there is then a location relation between objects and points or regions of spacetime (on another reading there is an object- or point-valued location determinable). Depending on the version of absolutism, the location relation may correspond to the predicate is wholly located at, is at least partly located at or is exactly located at (I may be leaving out some options).

Now the deep question is this: What is it that makes a relation between objects and points or regions of a topological space be a location relation? (The question can also be put on relationism. Then the question is what is it that makes a family of relations between objects be a family of spatial, or spatiotemporal, relations.)

There are two extreme answers.

Location monism: There is just one location relation. In a Newtonian and in an Einsteinian world and in a 12-dimensional discrete universe, one and the same relation relates objects to points or regions of a topological space, obviously a very different topological space in each case.

Location functionalism: Any natural (sufficiently natural? perfectly natural?) relation between objects and points in a topological space, where the topological space is either concrete and cosntituted as a topological space by natural relations, or abstract as in this post, is a location relation. What the axioms are will depend on which location relation one takes as fundamental as well as on difficult metaphysical issues. Supposing that the relation is being exactly located at, and the spatial relata are regions, then the axioms might be very lax. In fact they might be nothing but:

• If xLR and x is a part of y, then there is a unique region R' that contains R such that yLR'.
• If yLR and x is a part of y, then there is a unique region R' that is contained in R such that xLR'.

(If Thomistic part nihilism is true, do this with virtual parts.) On this view, a relation of being exactly located at any phase space with a topology will count as a location relation as long as the relation is in fact natural. One might additionally add some more axioms, such as that no object is exactly located at two distinct regions (though I myself am inclined to deny that as it's incompatible with my best account of transsubstantiation), but the result about phase spaces will remain true. If one wants to rule them, one can either insist that in fact there is no phase space location in which is natural or disallow abstract topological spaces as relata, despite the benefits of allowing them. One might also add an axiom that makes this be a location in spacetime, by using causation. For instance, we might require the topological space to have a partial ordering on its points (we might add something about how the ordering should play nice with the topology), which we will call "at least as late as", and then extend this to a relation between regions: R' is at least as late as R provided that every point y of R' is at least as late as some point x of R and every point x of R has some point y of R' such that y is at least as late as x. Then add:

• Normally, if event E causes event E', and E and E' are exactly located at R and R' respectively, then R' is at least as late as R.

Monism and functionalism are extreme theories because functionalism classifies as locational as many relations as anybody could possibly reasonably want to do that to and monism classifies as locational as few as anybody who thinks location is real reasonably could.

I incline to functionalism here.

The puzzle of change

There seems to be a difference between change and replacement. Sperm and egg unite, and are changed into a new organism. While the sperm and egg perish in being united, it seems that they do not simply perish, replaced by the new organism. Rather, they turn into the new organism in some sense. It is this "in some sense" that is puzzling here—what does it mean? It seems to mean more than just that the sperm and egg have a causal role in producing the new organism—that would be to reduce material causes to efficient causes, and would be unsatisfactory. If an artist created a sculpture and happened to cease to exist just as the sculpture was completed, we wouldn't say that the artist turned into the sculpture.

A traditional solution to the puzzle is to suppose that matter survives substantial change. The matter of the sperm and egg loses its old forms, and takes on a new form, that of the new organism. But the artist does not turn into the statue, because the statue is not made of the artist's matter. Now, on my preferred view of matter (and my reading of Metaphysics Z), matter does not survive substantial change.

So it seems that there is a good argument against my view of matter: if matter does not survive change, then one cannot distinguish change and replacement.

But I have a response. If there is in fact no difference between change and replacement, then the above is no objection to my view of matter. Suppose then there is a difference between change and replacement. Then we will want to say that in transsubstantiation the bread and wine are not replaced by, but changed into, the body and blood of our Savior. But the matter of bread and wine do not survive transsubstantiation. Hence it cannot be the case that the only way to distinguish between change and replacement is through the persistence of matter.