Three-dimensionality

It seems surprising that space is three-dimensional. Why so few dimensions?

An anthropic answer seems implausible. Anthropic considerations might explain why we don’t have one or two dimensions—perhaps it’s hard to have life in one or two dimensions, Planiverse notwithstanding—but thye don’t explain why don’t have thirty or a billion dimensions.

A simplicity answer has some hope. Maybe it’s hard to have life in one and two dimensions, and three dimensions is the lowest dimensionality in which life is easy. But normally when we do engage in simplicity arguments, mere counting of things of the same sort doesn’t matter much. If you have a theory on which in 2050 there will be 9.0 billion people, your theory doesn’t count as simpler in the relevant sense than a theory on which there will be 9.6 billion then. So why should counting of dimensions matter?

There is something especially mathematically lovely about three dimensions. Three-dimensional rotations are neatly representable by quaternions (just as two-dimensional ones are by complex numbers). There is a cross-product in three dimension (admittedly as well as in seven!). Maybe the three-dimensionality of the world suggests that it was made by a mathematician or for mathematicians? (But a certain kind of mathematician might prefer an infinite-dimensional space?)

The void between the atoms

Philoponus says:

When Democritus said that the atoms are in contact with each other, he did not mean contact, strictly speaking, which occurs when the surfaces of the things in contact fit on [epharmazousōn] one another, but the condition in which the atoms are near one another and not far apart is what he called contact. For no matter what, they are separated by void. (67A7)

This odd view would lead to three difficulties. First, the loveliness of the Democritean system is that everything is explained by atoms pushing each other around, without any mysterious action at a distance, without any weird forces like the love and strife posited by other Greek thinkers. But if two atoms are moving toward each other, and they must stop short of touching each other, it seems that we have some kind of a repulsion at a “near” distance. Second, the atomists thought everything happened of necessity. But why should two atoms heading for each other stop at distance x apart rather than distance x/2 or x/3, say? This seems arbitrary. And, third, what reason would Democritus have to say such a strange thing?

One solution is to simply say Philoponus was wrong about Democritus (cf. this interesting paper). One might, for instance, speculate that Democritus said something about how there will always be interstices of void when atoms meet, much like the triangle-like interstices when you tile the plane with circles in a hexagonal pattern, because their surfaces do not perfectly match like jigsaw pieces would, and Philoponus confused this with the claim that there is void between the atoms.

But I want to try something else. There is a famous problem—discussed by Sextus Empiricus, the Dalai Lama (!) and a number of people in between—about how impenetrable material objects can possibly touch. For if they touch, their surfaces are either separated by some distance or not.If their surfaces are separated, they don’t really touch. If their surfaces are not separated, then the surfaces are in the same place, and the objects have penetrated each other (albeit only infinitesimally) and hence they are not really impenetrable.

Suppose now that we think that Democritus was aware of this problem, and posited the following solution. Atoms occupy open regions of space, ones that do not include any of their boundaries or surfaces. For instance, atoms of fire, which are spherical, occupy the set of points in space whose distance to the center is strictly less than a radius r: the boundary, where the distance to the center is exactly r, is unoccupied. If two spherical atoms, each of radius r, come in contact, the distance between their centers is 2r, but the point exactly midway between their centers is not occupied by either atom. There is a single point’s worth of void there.

This immediately solves two of the three problems I gave for the void-between-atoms view. If I’m right, Democritus has very good reason to posit the view: it is needed to avoid the problem of interpenetration of surfaces. Furthermore, the arbitrariness problem disappears. Atoms heading for each other stop precisely when their boundaries would interpenetrate if they had boundaries in them. They stop at distance zero. There is no smaller distance they could stop at. The two spherical atoms stop moving toward each other when there is exactly one point of void between them: any more and they could keep on moving; any less is impossible.

We still have the problem of mysterious action at a distance requiring some force beyond mere contact. But Democritus might think—I don’t know if he would be right—that action at zero distance is less mysterious than action at positive distance, and on the suggestion I am offering the distance between objects that are touching is zero. There is a point’s (or a surface of points, if say we have two cubical atoms meeting with parallel faces) worth of distance, and that’s zero. Impenetrability at least explains why the atoms can’t go any further towards each other, even if it does not explain why they deflect each other’s motion as they do (which anyway, as we learn from Hume’s discussion of billiard balls, isn’t easy). So the remaining problem is reduced.

It wouldn’t surprise me at all if this was in the literature already.

Absolute reference frame

Some philosophers think that notwithstanding Special Relativity, there is a True Absolute Reference Frame. Suppose this is so. This reference frame, surely, is not our reference frame. We are on a spinning planet rotating around a sun orbiting the center of our galaxy. It seems pretty likely that if there is an absolute reference frame, then we are moving with respect to it at least at the speed of the flow of the Local Group of galaxies due to the mass of the Laniakea Supercluster of galaxies, i.e., at around 600 km/s.

Given this, our measurements of distance and time are actually going to be a little bit objectively off the true values, which are the ones that we would measure if we were in the absolute reference frame. The things we actually measure here in our solar system will be objectively off due to time dilation and space contraction by about two parts per million, if my calculations are right. That means that our best possible clocks will be objectively about a minute(!) off per year, and our best meter sticks will be about two microns off. Not that we would notice these things, since the absolute reference frame is not observable, so we can’t compare our measurements to it.

As a result, we have a choice between two counterintuitive claims. Either we say that duration and distance are relative, or we have to say that our best machining and time measuring is necessarily off, and we don’t know by how much, since we don’t know what the True Absolute Reference Frame is.

Reducing binary distance relations to unary properties

Some philosophers say that space is fundamentally constituted by points. Others that it is fundamentally constituted by regions, and points are logical constructions out of regions. Here is an interesting advantage of an approach base on regions. Relations are more mysterious that properties. A point-based account is likely to involve distance relations: x and y are α units apart.

But a region-based account need not suppose a distance relation, but a diameter property. Intuitively, the diameter of a region is the largest distance between two points in the region, and hence is defined in terms of a distance relation (to account for regions that are not compact, we need to say that the diameter is the supremum of the distances between points in the region). But we could also suppose that the diameter property is more fundamental than distance, and just as we might define points as constructions out of a region-based ontology, we might define distances as constructions out of diameters plus region mereology.

How this would work depends on the details of the point construction. One kind of point construction identifies points with (equivalence classes of) sequences of regions that get smaller and smaller. Some have done this with special concentric regions like balls, but one can also do it with more general regions making use of the diameter D(A) of a region A. Specifically, we can let a point be (an equivalence class of) a sequence of A₁, A₂, ... of regions, where we requires that later regions in the sequence always being subregions of the earlier ones, and that the limit of D(A_n) is zero.(The equivalence relation can be defined by stipulating that the sequences A₁, A₂, ... and B₁, B₂, ... are equivalent just in case D(A_n+B_n) converges to zero where A_n + B_n is the fusion of A_n and B_n.) We can then stipulate the distance between the points defined by the sequences A₁, A₂, ... and B₁, B₂, ... is equal to the limit of D(A_n+B_n). We’re going to need some axioms concerning diameters and regions for all this to be well-defined and for the distance to be a metric.

Or we can take a version of Lewis’s construction where points are just identified with balls of a specific diameter δ₀, with the intuition that we identify a point with the ball of diameter δ₀ "centered on it". And we can again define distances in terms of diameters: d(A,B) = D(A+B) − δ₀.

This does not rid us of all relations. After all, we are supposing the mereological parthood relation (in its "subregion" special case). However, one might think that parthood is more of a fundamental binary predicate than a relation. And at least it’s not a determinable relation, in the way that distance is.

I am not myself fond of mereology. So the above is not something I am going to push. But it would be fun to work out the needed axioms if nobody’s done it (quite likely someone has—maybe Lewis, as I haven’t actually read his stuff on this, but am going on hearsay). It would make a nice paper for a grad student who likes technical stuff.

Saving a Newtonian intuition

Here is a Newtonian intuition:

1. Space and time themselves are unaffected by the activities of spatiotemporal beings.

General Relativity seems to upend (1). If I move my hand, that changes the geometry of spacetime in the vicinity of my hand, since gravity is explained by the geometry of spacetime and my hand has gravity.

It’s occurred to me this morning that a branching spacetime framework can restore the Newtonian intuition of the invariance of space. Suppose we think of ourselves as inhabiting a branching spacetime, with the laws of nature being such as to require all the substances to travel together (cf. the traveling forms interpretation of quantum mechanics). Then we can take this branching spacetime to have a fixed geometry, but when I move my hand, I bring it about that we all (i.e., all spatiotemporal substances now existing) move up to a branch with one geometry rather than up to a branch with a different geometry.

On this picture, the branching spacetime we inhabit is largely empty, but one lonely red line is filled with substances. Instead of us shaping spacetime, we travel in it.

I don’t know if (1) is worth saving, though.

Does general relativity lead to non-locality all on its own?

A five kilogram object has the determinable mass with the determinate mass of 5 kg. The determinate mass of 5 kg is a property that is one among many determinate properties that together have a mathematical structure isomorphic to a subset of real numbers from 0 to infinity (both inclusive, I expect). Something similar is true for electric charge, except now we can have negative values. Human-visible color, on the other hand, lies in a three-dimensional space.

I think one can have a Platonic version of this theory, on which all the possible determinate properties exist, and an Aristotelian one on which there are no unexemplified properties. There will be important differences, but that is not what I am interested in in this post.

I find it an attractive idea that spatial location works the same way. In a Newtonian setting the idea would be that for a point particle (for simplicity) to occupy a location is just to have a determinate position property, and the determinate position properties have the mathematical structure of a subset of three-dimensional Euclidean space.

But there is an interesting challenge when one tries to extend this to the setting of general relativity. The obvious extension of the story is that determinate instantaneous particle position properties have the mathematical structure of a subset of a four-dimensional pseudo-Riemannian manifold. But which manifold? Here is the problem: The nature of the manifold—i.e., its metric—is affected by the movements of the particles. If I step forward rather than back, the difference in gravitational fields affects which mathematical manifold our spacetime is isomorphic to. If determinate position properties are tied to a particular manifold, it means that the position of any massive object affects which manifold all objects are in and have always been in. In other words, the account seems to yield a story that is massively non-local.

(Indeed, the story may even involve backwards causation. Since the manifold is four-dimensional, by stepping forward rather than backwards I affect which four-dimensional manifold is exemplified, and hence which manifold particles were in. )

This is interesting: it suggests that, on a certain picture of the metaphysics of location, general relativity by itself yields non-locality.

Property dualism and relativity theory

On property dualism, we are wholly made of matter but there are irreducible mental properties.

What material object fundamentally has the irreducible mental properties? There are two plausible candidates: the body and the brain. Both of them are extended objects. For concreteness, let’s say that the object is the brain (the issue I will raise will apply in either case) Because the properties are irreducible and are fundamentally had by the brain, they are are not derivative from more localized properties. Rather, the whole brain has these properties. We can think (to borrow a word from Dean Zimmerman) that the brain is suffused with these fundamental properties.

Suppose now that I have an irreducible mental property A. Then the brain as a whole is suffused with A. Suppose that at a later time, I cease to have A. Then the brain is no longer suffused with A. Moreover, because it is the brain as a whole that is a subject of mental properties, it seems to follow that the brain must instantly move from being suffused as a whole with A to having no A in it at all. Now, consider two spatially separated neurons, n₁ and n₂. Then at one time they are both participate in the A-suffusion and at a later time neither participates in the A-suffusion. There is no time at which n₁ (say) participates in A-suffusion but n₂ does not. For if that were to happen, then A would be had by a proper part of the brain then rather than by the brain as a whole, and we’ve said that mental properties are had by the brain as a whole.

But this violates Relativity Theory. For if in one reference frame, the A-suffusion leaves n₁ and n₂ simultaneously, then in another reference frame it will leave n₁ first and only later it will leave n₂.

I think the property dualist has two moves available. First, they can say that mental properties can be had by a proper part of a brain rather than the brain as a whole. But the argument can be repeated for the proper part in place of the brain. The only stopping point here would be for the property dualist to say that mental properties can be had by a single point particle, and indeed that when mental properties leave us, at some point in time in some reference frames they are only had by very small, functionally irrelevant bits of the brain, such as a single particle. This does not seem to do justice to the brain dependence intuitions that drive dualists to property dualism over substance dualism.

The second move is to say that the brain as a whole has the irreducible mental property, but to have it as a whole is not the same as to have its parts suffused with the property. Rather, the having of the property is not something that happens to the brain qua extended, spatial or composed of physical parts. Since physical time is indivisible from space, mental time will then presumably be different from physical time, much as I think is the case on substance dualism. The result is a view on which the brain becomes a more mysterious object, an object equipped with its own timeline independent of physics. And if what led people to property dualism over substance dualism was the mysteriousness of the soul, well here the mystery has returned.

Some fun distinctions

Isn’t it funny how very similar gestures can signal respect and disrespect? Under ordinary circumstances, crossing to the other side of the street to avoid near someone is a form of disrespect. But in a pandemic it signals a respectful desire not to make the other nervous. Though I suppose even apart from a pandemic, one would have moved out of the way of dignitaries.

We have another neat little thing here. There is a difference between going out of one’s way to ensure that one isn’t in another’s personal space and going out of one’s way to ensure that the other isn’t in one’s personal space, even though in an egalitarian society, x is in y’s space if and only if y is in x’s space.

And notice how hard it is to formulate that point without reifying “personal space”, just by using distance. I can hear a difference between avoiding my being within a certain distance of another and avoiding the other being within a certain distance of me, but I can’t tell which is which! Maybe, though, we can distinguish (a) avoiding imposing on another the bad-for-them of us being within a certain distance and (b) avoiding imposing on me the bad-for-me of us being within that distance. In other words, the reasons for the two actions are grounded in the same state of affairs but considered as bad for different individuals.

I suppose similar things can happen entirely in third person contexts. I can work for a friendship between x and y considered as a good for x, considered as a good for y, or considered as a good for both. And these are all three different actions.

Leaving space

Suppose that we are in an infinite Euclidean space, and that a rocket accelerates in such a way that in the first 30 minutes its speed doubles, in the next 15 minutes it doubles again, in the next 7.5 minutes it doubles, and so on. Then in each of the first 30 minutes, and the next 15 minutes, and the next 7.5 minutes, and so on, it travels roughly the same distance, and over the next hour it will have traveled an infinite distance. So where will it be? (This is a less compelling version of a paradox Josh Rasmussen once sent me. But it’s this version that interests me in this post.)

The causal finitist solution is that the story is impossible, for the final state of the rocket depends on infinitely many accelerations, and nothing can causally depend on infinitely many things.

But there is another curious solution that I’ve never heard applied to questions like this: after an hour, the rocket will be nowhere. It will exist, but it won’t be spatially related to anything outside of itself.

Would there be a spatial relationship between the parts of the rocket? That depends on whether the internal relationships between the parts of the rocket are dependent on global space, or can be maintained in a kind of “internal space”. One possibility is that all of the rocket’s particles would lose their spatiality and exist aspatially. Another is that they would maintain spatial relationships with each other, without any spatial relationships to things outside of the rocket.

While I embrace the causal finitist solution, it seems to me that the aspatial solution is pretty good. A lot of people have the intuition that material objects cannot continue to exist without being in space. I don’t see why not. One might, of course, think that spatiality is definitive of materiality. But why couldn’t a material object then continue to exist after having lost its materiality?

A new argument for presentism

Here’s an interesting argument favoring presentism that I’ve never seen before:

1. Obviously, a being that fails to exist at some time t is not a necessary being.

2. If presentism is true, we have an elegant explanation of (1): If x fails to exist at t₁, then at t₁ it is true that x does not exist simpliciter, and whatever is true at any time is possibly true, so it is possible that x does not exist simpliciter, and hence x is not a necessary being.

3. If presentism is false, we have no equally good explanation of (1).

4. So, (1) is evidence for presentism.

I don’t know how strong this argument is, but it does present an interesting explanatory puzzle for the eternalist:

5. Why is it that non-existence at a time entails not being necessary?

Here’s my best response to the argument. Consider the spatial parallel to (1):

6. Obviously, a being that fails to exist at some location z is not a necessary being.

It may be true that a being that fails to exist at some location is not a necessary being, since in fact the necessary being is God and God is omnipresent. But even if it’s true, it’s not obvious. If Platonism were true, then numbers would be counterexamples to (6), in that they would be necessary beings that aren’t omnipresent.

But numbers seem to be not only aspatial but also atemporal. And if that’s right, then (1) isn’t obvious either. (In fact, if numbers are atemporal, then they are a counterexample to presentism, since they don’t exist presently but still exist simpliciter.)

What if the presentist insists that numbers would exist at every time but would not be spatial? Well, that may be: but it’s far from obvious.

What if we drop the “Obviously” in (1)? Then I think the eternalist theist can give an explanation of (1): The only necessary being is God, and by omnipresence there is no time at which God isn’t present.

Maybe one can use the above considerations to offer some sort of an argument for presentism-or-theism.

Real Presence and primitive locational relations

According to relationalism, space is constituted by the network of spatial relations, such as metric distance relations (e.g., being seven meters apart). If these relations are primitive, then there is a very easy way for God to ensure the Real Presence of Christ: he can simply make there be additional spatial relations between Christ and other material entities, spatial relations that are exactly like the relations that the bread and wine stood in to other material entities.

It might seem contradictory for Christ to stand in two distance relations: for instance, being one mile from me (in one church) and three miles from me (in another). But I doubt this is a contradiction. New York and London are both 5600 km and 34500 km apart, depending on which direction you go.

According to substantivalism, on the other hand, points or regions are real, and objects are in a location by standing in a relation to a point or region. If relations are primitive, again there should be no problem about God instituting additional such relations to make it be that Christ is present where the bread and wine were.

In other words, if location is constituted by a primitive relation—whether to other objects or to space—there is apt to be no difficulty in accounting for the Real Presence. The reason is that we expect, barring strong reason to the contrary, primitive relations to be arbitrarily recombinable.

If location, however, is constituted by a non-primitive relation, there might be more difficulties. For instance, as a toy theory, consider the variant of relationalism on which spatial relations are constituted by gravitational force relations (two objects have distance r if and only if they have masses m₁ and m₂ and there is a gravitational force Gm₁m₂/r² between them). In that case, for God to make Christ present in Waco would require God to make Christ stand in gravitational force relations of the sort that I stand in by virtue of being in Waco. For instance, the earth’s gravitational force on Christ would have to point from Waco to the center of the earth—but since the Eucharist is also in Rome, it would have to point from Rome to the center of the earth as well. And that might be thought impossible. But perhaps there could be two terrestrial gravitational forces on Christ: one along the Waco-geocenter vector and the other along the Rome-geocenter vector. This would require some sort of a realism about component forces, but that’s probably necessary for the gravitational toy theory. And then God would have to miraculously ensure that despite the forces, Christ is not affected in the way he would normally be by these forces. All this may be possible, but it’s less clear than if we have primitive relations.

Holes and substantivalism

Suppose substantivalism about space is correct. Imagine now that the following happens to a slice of swiss cheese: the space where the holes were suddenly disappears. I don’t mean that the holes close up. I mean that the space disappears: all the points and regions that used to be in the hole are no longer there (and any air that used to be there is annihilated). The surfaces of the cheese that faced the hole now are at an edge of space itself.

The puzzle now is that in this story we have an inconsistent triad:

1. There is no intrinsic change in the cheese.
2. The slice of cheese no longer has holes.
3. Changing with respect to whether you have holes is intrinsic.

Here are my arguments for the three claims. There is no intrinsic change in the slice of cheese as something outside the cheese has changed—space has been annihilated. The slice of cheese no longer has holes, as it makes sense to talk of the size or shape or volume of a hole, but there is no size or shape or volume where there is no space. And changing with respect to whether you have holes is change of shape, and changes of shape are intrinsic.
It seems that the above story forces you to reject one of the following:

4. Substantivalism about space
5. Intrinsicness of shape.

But there is another way out. Deny (3). Whether you have holes is not intrinsic. What is intrinsic is your topological genus with respect to your internal space and similar topological properties.

Note, also, a lesson relevant to the famous Lewis and Lewis paper on holes: the counting of holes should not involve the counting of regions, but the computation of a numerical invariant, namely the genus.

Internal reference frame

Suppose a long snake is stretched out and its front half is annihilated instantaneously. This presumably instantly destroys the snake's form or soul. So the tip of the snake's tail instantly ceases to be informed by the snake form. But then there will be a reference frame according to which the front half is annihilated before the tail loses its form. In that frame, the tip of the tail still has a snake form at a time at which the snake's front half doesn't exist. That seems wrong. So it seems there should be a privilege to reference frames where the front half is destroyed simultaneously with the tail losing its form. But a global privileged frame is unattractive. Maybe, however, we should suppose that particular substances carry along privileged frames of their own, frames internal to them. Then there will be a privilege frame for each substance, but these frames need not cohere into a global privileged frame.

Velocity and teleportation

Suppose a rock is flying through the air northward, and God miraculously and instantaneously teleports the rock, without changing any of its intrinsic properties other than perhaps position, one meter to the west. Will the rock continue flying northward due to inertia?

If velocity is defined as the rate of change of position, then no. For the rate of change of position is now westward and the magnitude is one meter divided by zero seconds, i.e., infinite. So we cannot expect inertia to propel the rock northward any more. In fact, at this point physics would break down, since the motion of an object with infinite velocity cannot be predicted.

But if velocity (or perhaps momentum) is an intrinsic feature that is logically independent of position, and it is merely a law of physics that the rate of change of position equals the velocity, then even after the miraculous teleportation, the rock will have a northward velocity, and hence by inertia will continue moving northward.

I find the second option to be the more intuitive one. Here is an argument for it. In the ordinary course of physics, the causal impact of physical events at times prior to t₁ on physical events after t₁ is fully mediated by the physical state of things at t₁. Hence whether an object moves after time t₁ must depend on its state at t₁, and only indirectly on its state prior to t₁. But if velocity is the rate of change of position, then whether an object moves via inertia after t₁ would depend on the position of the object prior to t₁ as well as at t₁. So velocity is not the rate of change of position, but rather a quality that it makes sense to attribute to an object just in virtue of how it is at one time.

This would have the very interesting consequence that it is logically possible for an object to have non-zero velocity while not moving: God could just constantly prevent it from moving without changing its velocity.

An argument that motion doesn't supervene on positions at times

In yesterday’s post, I offered an argument by my son that multilocation is incompatible with the at-at theory of motion. Today, I want to offer an argument for a stronger conclusion: multilocation shows that motion does not even supervene on the positions of objects at times. In other words, there are two possible worlds with the same positions of objects at all times, in one of which there is motion and in the other there isn’t.

The argument has two versions. The first supposes that space and time are discrete, which certainly seems to be logically possible. Imagine a world w₁ where space is a two-dimensional grid, labeled with coordinates (x, y) where x and y are integers. Suppose there is only one object, a particle quadlocated at the points (0, 0), (1, 0), (0, 1) and (1, 1). These points define a square. Suppose that for all time, the particle, in all its four locations, continually moves around the square, one spatial step at a temporal step, in this pattern:

(0, 0)→(1, 0)→(1, 1)→(0, 1)→(0, 0).
Then at every moment of time the particle is located at the same four grid points. But it is also moving all the time.

But there is a very similar world, w₂, with the same grid and the same multilocated particle at the same four grid points, but where the particle doesn’t move. The positions of all the objects at all the times in w₁ and w₂ are the same, but w₁ has motion and w₂ does not.

Suppose you don’t think space and time can be discrete. Then I have another example, but it involves infinite multilocation. Imagine a world w₃ where the universe contains a circular clock face plus a particle X. None of the particles making up the clock face move. But the particle X uniformly moves clockwise around the edge of the clock face, taking 12 hours to do the full circle. Suppose, further, that X is infinitely multilocated, so that it is located at every point of the edge of the clock face. In all its locations X moves around the circle. Then at every moment of time the particle is located at the same point, and yet it is moving all the time.

Now imagine a very similar world w₄ with the same unmoving clock face and the same spacetime, but where the particle X is eternally still at every point on the edge of the clock face. Then w₃ and w₄ have the same object positions at all times, but there is motion in w₃ and not in w₄.

I think the at-at theorist’s best bet is just to deny that there is any difference between w₁ and w₂ or between w₃ and w₄. That’s a big bullet to bite, I think.

It would be nice if there were some way of adding causation to the at-at story to solve these problems. Maybe this observation would help: When the particle in w₁ moves from (0, 0) to (1, 0), maybe this has to be because something exercises a causal power to make a particle that was at (0, 0) be at (1, 0). But there is no such exercise of a causal power in w₂.

External time as such doesn't matter to us

Suppose a deity threatened to move us all to a universe where everything is pretty much as in our world, except that electric charges are reversed and the laws of nature are tweaked to ensure that this reverse doesn’t affect our lives. Thus, in that world, we are based not on carbon atoms, but on anti-carbon anti-atoms (they will have six anti-protons and six positrons, etc.), but the laws are tweaked so that the anti-atoms would behave just like atoms.

Assuming we can survive the shift, it seems that except for sentimental considerations (maybe when Grandma’s old wedding ring is replaced by a ring of anti-gold, it’s no longer the same ring) it would make no difference to us.

Similarly, if the deity threatened to spatially rotate the world by 180 degrees around some axis, that would make no difference to us.

What if the deity offered to rotate our world in time by 180 degrees, with causation now running temporally backwards, with us being born in the future and dying in the past, but everything being kept intact. It seems to me that this would make no difference to us.

Similarly, it seems to me that if the deity offered to rotate our four-dimensional world so that the temporal dimension and a spatial dimension were swapped, so that we would be born and die at the same time, but in different places along a spatial axis, and causation would run unidirectionally along the spatial axis, again that would make no difference to us.

I think these thought experiments suggest that external time as such is not important. What matters is how the distribution of things interacts with the causal order.

To be honest, though, I am not completely confident that any of these thought experiments make sense. It could be that any dimension along which causation runs much as causation runs along the temporal direction in our world is therefore a time dimension. But if so, then I think it's still true that it is causation, not external time as such, that matters.

I am less confident of this in the case of internal time.

Are monads in space?

It is often said that Leibniz’s monads do not literally occupy positions in space. This seems to me to be a mistake, perhaps a mistake Leibniz himself made. Leibnizian space is constituted by the perceptual relations between monads. But if that’s what space is, then the monads do occupy it, because they stand in the perceptual relations that constitute space. And they occupy it literally. There is no other way to occupy space, if Leibniz is right: this is literal occupation of space.

Perhaps the reason it is said that the monads do not literally occupy positions in space is that an account that reduces position to mental properties seems to be a non-realist account of position. This is a bit strange. Suppose we reduce position to gravitational force and mass (“if objects have masses m₁ and m₂ and a gravitational force F between them, then their distance is nothing but (Gm₁m₂/F)^1/2”). That’s a weird theory, but a realist one. Why, then, should a reduction to mental properties not be a realist one?

Maybe that’s just definitional: a reduction of physical properties to mental ones counts as a non-realism about the physical properties. Still, that’s kind of weird. First, a reduction of mental properties to physical ones doesn’t count as a non-realism about the mental properties. Second, a reduction of some mental properties to other mental properties—say, beliefs to credence assignments—does not count as non-realism about the former. Why, then, is a reduction of physical to mental properties count as a non-realism?

Maybe it is this thought. It seems to be non-realist to reduce some properties to our mental properties, where “our” denotes some small subset of the beings we intuitively think exist. Thus, it seems to be non-realist to reduce aesthetic properties to the desires and beliefs of persons, or to reduce stones to the perceptual properties of animals. But suppose we are panpsychist as Leibniz is, and think there are roughly at least as many beings as we intuitively think there are, and are reducing physical properties to the mental properties of all the beings. Then it’s not clear to me that that is any kind of non-realism.

Extended simples

1. It is possible to have a simple that exists at more than one time.

2. Four-dimensionalism is true.

3. So, temporally extended simples are possible. (By 1 and 2)

4. If four-dimensionalism is true, the time and space are metaphysically very similar.

5. So, probably, spatially extended simples are possible.

Inductive evidence of the existence of non-spatial things

Think about other plausibly fundamental qualities beyond location and extension: thought, charge, mass, etc. For each one of these, there are things that have it and things that don’t have it. So we have some inductive reason to think that there are things that have location and things that don’t, things that have extension and things that don’t. Admittedly, the evidence is probably pretty weak.

A spatial "in between"

In my last post I offered the suggestion that someone who thinks time is discrete has reason to think that there is something in between the moments—a continuous unbroken (but perhaps breakable) interval.

I think a similar thought can be had about discrete space.

Consideration 1: Imagine that space is discrete, arranged on a grid pattern, and I touch left and right index fingers together. It could happen that the rightmost spatial points of my left fingertip is side-by-side with the leftmost spatial points of my right fingertip, but nonetheless my hands aren’t joined into a single solid. One way to represent this setup would be to say that a spatial point in my left fingertip is right next to a spatial point in my right fingertip, but the interval between these spatial points is not within me.

But positing a spatial “in between” isn’t the only solution: distinguishing internal and external geometry is another.

Consideration 2: Zeno’s Stadium argument can be read as noting that if space and time are discrete, then an object moving at one point per unit of time rightward and an equal length object moving at one point per unit of time leftward can pass by each other without ever being side-by-side. Positing an “in between”, such that objects may be “inbetween places when they are in between times, may make this less problematic.