More on Newtonian velocity

Here is a big picture story about Newtonian mechanics: The state of the system at all times t > t₀ is explained by the initial conditions of the system at t₀ and the prevalent forces.

But what are the initial conditions? They include position and velocity. But now here is a problem. The standard definition of velocity is that it is the time-derivative of position. But the time-derivative of position at t₀ logically depends not just on the position at t₀ but also on the position at nearby times earlier and later than t₀. That means that the evolution of the system at times t > t₀ is explained by data that includes information on the state of the system at times later than t₀. This seems explanatorily circular and unacceptable.

There is an easy mathematical fix for this. Instead of defining the velocity as the time-derivative position, we define the velocity as the left time-derivative of position: v(t)=lim_{h → 0−}(x(t + h)−x(t))/h. Now the initial conditions at t₀ logically depend only on what happens at t₀ and at earlier times.

This fixed Newtonian story still has a serious problem. Suppose that the system is created at time t₀ so there are no earlier times. The time-derivative at t₀ is then undefined, there is no velocity at t₀, and Newtonian evolution cannot be explained any more.

Here’s another, more abstruse, problem with the fixed Newtonian story. Suppose I am in a region of space with no forces, and I have been sitting for an hour preceding noon in the same place. Then at noon God teleports me two meters to the right along the x-axis, so that at all times before noon my position is x₀ and at noon it is x₀ + 2. Suppose, further, that the teleportation is the only miracle God does. God doesn’t change any other properties of me besides position, and God lets nature take over at all times after noon.

What will happen to me after noon? Well, on the fixed Newtonian story, my velocity at noon is the left-derivative of position, i.e., lim_{h → 0−}(2 − 0)/(0 − h)= + ∞. Since there are no prevailing forces, my acceleration is zero, and so my velocity stays unchanged. Hence, at all times after noon, I have infinite velocity along the x-axis, and so at all times after noon I end up at distance infinity from where I was—which seems to make no sense at all!

So the left-derivative fix of the Newtonian story doesn’t seem right, either, at least in this miracle case.

My preference to both the original Newtonian story and the fixed story is to take velocity (or perhaps momentum) to be a fundamental physical quantity that is not defined as the derivative, or even left derivative, of position.

The rest is technicalities. Maybe we could now take Newton’s Second Law to be:

1. ∂_t⁺v(t)=F/m,

where ∂_t⁺ is the right (!) time-derivative, and add two new laws of nature:

2. ∂_t⁺x(t)=v(t), and

3. x(t) and v(t) are both left (!) continuous.

Now, (2) is an explicit law of nature about the interaction of velocity and position rather than a definition of velocity. On this picture, here’s what happens in the teleportation case. Before noon, my velocity is zero and my position is x₀. Because I supposed that the only thing that God miraculously affects is my position, my velocity is still zero at noon, even though my position is now x₀ + 2. And I think (by the answer to this), laws (1), (2) and (3) ensure that if there are no further miracles, I remain at x₀ + 2 in the absence of external forces. The miraculous teleportation violates (2) and (3) at noon and at no other times.

But of course this is all on the false premise of Newtonian mechanics.

A philosophical advantage of quantum mechanics over Newtonian mechanics

We often talk as if quantum mechanics were philosophically much more puzzling than classical mechanics. But there is also a deep philosophical puzzle about Newtonian mechanics as originally formulated—the puzzle of velocities—which disappears on quantum mechanics.

The puzzle of velocities is this. To give a causal explanation of a Newtonian system’s behavior, we have to give the initial conditions for that system. These initial conditions have to include the positions and velocities (or momenta) of all the bodies in the system.

To see why this is puzzling, let’s imagine that t₀ is the first moment of the universe’s existence. Then the conditions at t₀ explain how things are at all times t > t₀. But how can there be velocities at t₀? A velocity is a rate of change of position over time. But if t₀ is the first moment of the universe’s existence, there were no earlier positions. Granted, there are later positions. But these later positions, given Newtonian dynamics, depend on the velocities at t₀ and hence cannot help determine what these velocities are.

One might try to solve this by saying that Newtonian dynamics implies that there cannot be a first moment of physical reality, that physical reality has to have always existed or that it exists on an interval of times open at the lower end. On either option, then, Newtonian dynamics would have to be committed to an infinite temporal regress, and that seems implausible.

Another solution would be to make velocities (or, more elegantly, momenta) equally primitive with positions (indeed, some mathematical formulations will do that). On this view, that the velocity is the rate of change of position would no longer be a definition but a law of nature. This increases the number of laws of nature and the fundamental properties of things. And if it is a mere law of nature that velocity is the rate of change of position, then it would be metaphysically possible, by a miracle, that an object standing perfectly still for days would nonetheless have a high velocity. If that seems wrong, we could just introduce a technical term, say “movement propensity” (that’s kind of what “momentum” is), in place of “velocity”, and it would sound better. However, anyway, while the resulting theory would be mathematically equivalent to Newton’s, and it would solve the velocity problem, it would be a metaphysically different theory, since it would have different fundamental properties.

On the other hand, the whole problem is absent in quantum mechanics. The Schroedinger equation determines the values of the wavefunction at times later than t₀ simply on the basis of the values of the wavefunction at t₀. Granted, the cost is that we have a wavefunction instead of just positions. And in a way it is really a variant of the making-momenta-primitive solution to the Newtonian problem, because the wavefunction encodes all the information on positions and momenta.

The present doesn't ground the past

I will run an argument against the thesis that facts about the past are grounded in the present on the basis of the intuition that that would be a problematically backwards explanation.

Suppose for a reductio:

1. Necessarily, facts about the past are fully grounded in facts about the present.

Add the plausible premises:

2. Necessarily, if fact C is fully grounded in some facts, the Bs, and the Bs are fully causally explained by fact A, then fact A causally explains fact C.

As an illustration, suppose that the full causal explanation of why the Nobel committee gave the Nobel prize to Bob is that Alice persuaded them to. Bob’s being a Nobel prize winner is fully grounded in his being awarded the Nobel prize by the Nobel committee. So, Alice’s persuasion fully causally explains why Bob is the Nobel prize winner.

3. It is possible to have a Newtonian world such that:

   1. All the facts about the world at any one time are fully causally explained by the complete state of the universe at any earlier time.

   2. There are no temporally backwards causal explanations.

   3. There are at least three times.

Now, consider such a Newtonian world, and let t₁ < t₂ < t₃ be three times (by (3c)).

Suppose that t₃ is now present. Let U_i be the fact that the complete state of the universe at time t_i is (or will be or was) as it is (or will be or was). Then:

4. Fact U₁ is fully grounded in some facts about the present. (By (1))

Call these facts the Bs.

5. The Bs are fully causally explained by U₂. (As (3a) holds in our assumed world)

Therefore:

6. Fact U₁ is fully causally explained by U₂. (By (1))

7. So, there is backwards causal explanation. (By (6))

8. Contradiction! (By (7) and as (3b) holds in our assumed world)

I think we should reject (1), and either opt for eternalism or for Merricks’ version of presentism on which facts about the past are ungrounded.

A ray of Newtonian particles

Imagine a Newtonian universe consisting of an infinite number of equal masses equidistantly arranged at rest along a ray pointing to the right. Each mass other than first will experience a smaller gravitational force to the left and a greater (but still finite, as it turns out) gravitational force to the right. As a result, the whole ray of masses will shift to the right, but getting compressed as the masses further out will experience less of a disparity between the left-ward and right-ward forces. There is something intuitively bizarre about a whole collection of particles starting to move in one direction under the influence of their mutual gravitational forces. It sure looks like a violation of conservation of momentum. Not that such oddities should surprise us in infinitary Newtonian scenarios.

Instantaneous Newtonian gravitational causation at a distance?

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

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

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

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

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

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

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

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

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

The Tammes problem

I wanted to 3D print a ball with dimples like a golf ball, so I got to looking up how to evenly distribute points over the surface of a sphere. Thinking about this problem leads to a very natural optimization problem: given a natural number n, place n points on the surface of a sphere in a way that maximizes the shortest distance between any two points. This problem has a name: it is the Tammes problem. Of course, for my purposes, it really doesn't matter whether I have an exact solution to the problem--an approximate one will do.

A natural way to try to approximately solve the problem is to pretend that the points are particles that have repulsive forces between them, and then run a computer simulation of initially randomly distributed particles moving under the influence of these forces, with some frictional damping.

Inspired by this paper, I initially worked with a repulsive force inversely proportional to d^p, where d is the distance between the points and p is an exponent that is ramped up as the simulation progresses.

Experimenting with various parameters, I found it was helpful to start with p=1 and go up to p=4.5 and then stay at p=4.5 for a while before finishing the simulation. Velocity-dependent friction seems to work a little better than velocity-independent friction. The physical precision of the simulation, of course, doesn't matter at all, except as a means to getting a large minimum spacing. For 500 points, with 1000 steps of Euler-Cromer simulation and carefully tuned parameters (friction, dynamic step size, ramping schedule), I was able to get a minimum spacing of about 0.153 in one run. There is a theorem by Fejes-Tóth that implies one can't do better than 0.1702, so it's pretty close.

A hint from the above-linked paper helped along the way: one can arrange the initial positions of the particles to be symmetric around the origin (i.e., if we place one particle is at x, we place another at −x). Then we only need to simulate the motions of half of the particles, since the movements of the other half are just a reflection about the origin. Of course, this optimization only works if n is even (though if n is odd, it still may be worth arranging all but one particles symmetrically initially).

Then I had another idea. At each simulation time, we already calculate the current distance d_min between the two particles closest together, and what we want to do is to particularly strongly push apart those particles whose distance is close to that. After a fair amount of fine-tuning, I ended up modifying the repulsive force to (d−cd_min)^p, where now we ramp p from 1 to 4.5, and c from 0 to 0.9. The result was noticeably better: my best answer for n=500 went from 0.153 to about 0.162, and typical runs give me about 0.161 after only 500 steps.

In the videos, the diameters of the red ball are equal to d_min, so the point is to maximize the size of the balls without allowing them to collide. The code is written in C. It's been some time since I've programmed in C, so it was fun to go back to C. And it was also fun to go back to programming a numerical simulation, which I did a lot of back when I was a teenager. Since my teenage years, things have changed. Computers are so much faster that my ordinary laptop has no difficulty with handling n-body interaction for n around 500 or 1000: back as a teenager, the most I worked with was about n=64. Moreover, my laptop has multiple cores, and OpenMP makes it super easy to split n-body problems between cores. My Dell laptop does a 500 step calculation with 500 particles in 1.4 seconds (but for even n, we have a symmetry optimization that cuts computation time in about a half; 499 particles takes 2.6 seconds).

Enjoy the code. It's in messy but portable C, and there is a a 32-bit Windows binary that uses all the cores you have. Just give tammes one argument: the number of particles. Visualization is done by feeding an -animate option and piping to a Classic VPython script.

You can can generate an OpenSCAD golfball from the code by using a -scad option instead. Unfortunately, OpenSCAD is really slow in processing the output of tammes. The golfball on the right has n=336. I seem to have read that that's a pretty normal n for golfballs.

An Aristotelian argument for a causal principle

Start with these assumptions:

1. Laws of nature are grounded in the powers of things. (I.e., Aristotelian picture of laws.)
2. Space can be infinite.
3. Newtonian physics is metaphysically possible.

There is a somewhat handwaving argument that if (1)-(3) are true, then an object cannot come into existence ex nihilo for no cause at all, and hence we have a causal princople.

Here's why. Say that a gridpoint in a Newtonian three dimensional space is a point with coordinates (x,y,z) where x,y and z are integers (in some fixed unit system).

Given (1)-(3) and assuming that objects can pop into existence ex nihilo, it should possible to start with a universe of finite total mass and then for a Newtonian particle of equal non-zero mass to simultaneously pop into existence at all and only those gridpoints (x,y,z) where z is positive, with nothing popping into existence elsewhere. Here's why. At each gridpoint, the object should be able to pop into existence. But objects that pop into existence causelessly at one location in space would be doing so in complete oblivion of what happens at other gridpoints. There should be total logical independence between all the poppings into existence. If so, then any combination of poppings or non-poppings should be able to happen at the gridpoints, and in particular, it should possible to have particles of equal mass pop into existence at the gridpoints with positive z-coordinates but nowhere else. But if this happened, then each particle would experience an infinite force in the direction of the z-axis (this follows from Newton's shell theorem and some approximation work), which would result in an infinite acceleration, which is absurd.

A relativistic version of this argument would require that spacetime can be infinite, so we could arrange the particles popping into existence along a single backwards light-cone.

There is a more general point here. The above example will remind regular readers of an argument I recently gave for causal finitism. I think many paradox-based arguments for causal finitism can be turned into arguments for causal principles in something like the above way. If this is right, this is very cool, because we can get both premises of a Kalaam Cosmological Argument out of the paradoxes then.

Two ways of violating causal finitism

Causal finitism holds that the causal history of anything is finite. On purely formal grounds (and assuming the Axiom of Choice--or at least Dependent Choice), it turns out that there are exactly two ways that a world could violate causal finitism:

1. The world contains an infinite regress.
2. Some effect is caused by infinitely many causes.

Historically, a lot of attention has been paid to the first option, with arguments back and forth on whether this is possible. Not much attention has been paid to the second. But notice that in an infinite universe with Newtonian physics, we do have a type (2) violation of causal finitism, in that the motion of each object is instantaneously gravitationally caused by the pull of infinitely many objects. I suppose that insofar as that sort of a world seems logically possible, that's an argument against causal finitism, though not a decisive one.

But perhaps the last observation can be turned into an argument for causal finitism. For if it is possible to have infinitely many objects working together causally, it should be possible to haven an infinite Newtonian universe. But it would be strange to suppose that some but not all infinite arrangements of physical objects are compossible with the Newtonian laws. After all, we can imagine asking: "What would happen if angels shuffled stuff?" So it should be possible to suppose a universe that has nothing in the half to the left of me, but in the half of the universe to my right is an infinite number of objects arranged in a uniform density in space. If that happened, I would experience an infinite force to the right (think of the gravitational force of a solid ball of uniform density at the surface: the Newtonian law makes the force be proportional to the ball's radius as the cube-dependence of the mass beats out the inverse-square-dependence), and accelerate infinitely to the right. That's impossible.

Zeno's arrow, Newtonian mechanics and velocity

Start with Zeno's paradox of the arrow. Zeno notes that over every instant of time t₀, an arrow occupies one and the same spatial location. But an object that occupies one and the same spatial location over a time is not moving at that time. (One might want to refine this to handle a spinning sphere, but that's an exercise to the reader.) So the arrow is not moving at t₀. But the same argument applies to every time, so the arrow is not moving, indeed cannot move.

Here's a way to, ahem, sharpen The Arrow. Suppose in our world we have an arrow moving at t₀. Imagine a world w* where the arrow comes into existence at time t₀, in exactly the same state as it actually has at t₀, and ceases to exist right after t₀. At w* the arrow only ever occupies one position—the one it has at t₀. Something that only ever occupies one position never moves (subject to refinements about spinning spheres and the like). So at w* the arrow never moves, and in particular doesn't move at t₀. But in the actual world, the arrow is in the same state at t₀ as it is at w* at that time. So in the actual world, the arrow doesn't move at t₀.

A pretty standard response to The Arrow is that movement is not a function of how an object is at any particular time, it is a function of how, and more precisely where, an object is at multiple times. The velocity of an object at t₀ is the limit of (x(t₀+h)−x(t))/h as h goes to zero, where x(t) is the position at t, and hence the velocity at t₀ depends on both x(t₀) and on x(t₀+h) for small h.

Now consider a problem involving Newtonian mechanics. Suppose, contrary to fact, that Newtonian physics is correct.

Then how an object will behave at times t>t₀ depends on both the object's position at t₀ and on the object's velocity at t₀. This is basically because of inertia. The forces give rise to a change in velocity, i.e., the acceleration, rather than directly to a change in position: F(t)=dv(t)/dt.

Now here is the puzzle. Start with this plausible thought about how the past affects the future: it does so by means of the present as an intermediary. The Cold War continues to affect geopolitics tomorrow. How? Not by reaching out from the past across a temporal gap, but simply by means of our present memories of the Cold War and the present effects of it. This is a version of the Markov property: how a process will behave in the future depends solely on how it is now. Thus, it seems:

1. What happens at times after t₀ depends on what happens at time t₀, and only depends on what happens at times prior to t₀ by the mediation of what happens at time t₀.

But on Newtonian mechanics, how an object will move after time t₀ depends on its velocity at t₀. This velocity is defined in terms of where the object is at t₀ and where it is at times close to t₀. An initial problem is that it also depends on where the object is at times later than t₀. This problem can be removed. We can define the velocity here solely in terms of times less than t₀, as lim_h→0−(x(t+h)−x(t))/h, i.e., where we take the limit only over negative values of h.[note 1] But it still remains the case that the velocity at t₀ is defined in terms of where the object is at times prior to t₀, and so how the obejct wil behave at times after t₀ depends on what happens at times prior t₀ and not just on what happens at t₀, contrary to (1).

Here's another way to put the puzzle. Imagine that God creates a Newtonian world that starts at t₀. Then in order that the mechanics of the world get off the ground, the objects in the world must have a velocity at t₀. But any velocity they have at t₀ could only depend on how the world is after t₀, and that just won't do.

Here is a potential move. Take both position and velocity to be fundamental quantities. Then how an object behaves after time t₀ depends on the object's fundamental properties at t₀, including its velocity then. The fact that v(t₀)=lim_h→0(x(t₀+h)−x(t₀))/h, at least at times t₀ not on the boundary of the time sequence, now becomes a law of nature rather than definitional.

But this reneges on our solution to The Arrow. The point of that solution was that velocity is not just a matter of how an object is at one time. Here's one way to make the problematic nature of the present suggestion vivid, along the lines of my Sharpened Arrow. Suppose that the arrow is moving at t₀ with non-zero velocity. Imagine a world w* just like ours at t₀ but does not have any times other than t₀.[note 2] Then the arrow has a non-zero velocity at t₀ at w*, even though it is always at exactly the same position. And that sure seems absurd.

The more physically informed reader may have been tempted to scoff a bit as I talked of velocity as fundamental. Of course, there is a standard move in the close vicinity of the one I made, and that is not to take velocity as fundamental, but to take momentum as fundamental. If we make that move, then we can take it to be a matter of physical law that mlim_h→0(x(t₀+h)−x(t₀))/h=p(t₀), where p(t) is the momentum at t.

We still need to embrace the conclusion that an object could fail to ever move and yet at have a momentum (the conclusion comes from arguments like the Sharpened Arrow). But perhaps this conclusion only seems absurd to us non-physicists because we were early on in our education told that momentum is mass times velocity as if that were a definition. But that is definitely not a definition in quantum mechanics. On the suggestion that in Newtonian mechanics we take momentum as fundamental, a suggestion that some formalisms accept, we really should take the fact that momentum is the product of mass and velocity (where velocity is defined in terms of position) to be a law of nature, or a consequence of a law of nature, rather than a definitional truth.

Still, the down-side of this way of proceeding is that we had to multiply fundamental quantities—instead of just position being fundamental, now position and momentum are—and add a new law of nature, namely that momentum is the product of mass and velocity (i.e., of mass and the rate of change of position).

I think something is to be said for a different solution, and that is to reject (1). Then momentum can be a defined quantity—the product of mass and velocity. Granted, the dynamics now has non-Markovian cross-time dependencies. But that's fine. (I have a feeling that this move is a little more friendly to eternalism than to presentism.) If we take this route, then we have another reason to embrace Norton's conclusion that Newtonian mechanics is not always deterministic. For if a Newtonian world had a beginning time t₀, as in the example involving God creating a Newtonian world, then how the world is at and prior to t₀ will not determine how the world will behave at later times. God would have to bring about the initial movements of the objects, and not just the initial state as such.

Of course, this may all kind of seem to be a silly exercise, since Newtonian physics is false. But it is interesting to think what it would be like if Newtonian physics were true. Moreover, if there are possible worlds where Newtonian physics is true, the above line of thought might be thought to give one some reason to think that (1) is not a necessary truth, and hence give one some reason to think that there could be causation across temporal gaps, which is an interesting and substantive conclusion. Furthermore, the above line of thought also shows how even without thinking about formalisms like Hamiltonian mechanics one might be motivated to take momentum to be a fundamental quantity.

And so Zeno's Arrow continues to be interesting.

More about functionalism about location

Functionalism about location holds that any sufficiently natural relation, say between objects and points in a topological space, that has the right formal properties (and, maybe, interacts the right way with causation) is a location relation.

Here is an argument against functionalism. Functionalism is false for other fundamental physical determinables: it is false for mass, charge, charm, etc. There is a possible world where some force other than electromagnetic is based on a determinable other than charge, but where the force and determinable follow structurally the same laws. By induction, functionalism is probably false for location.

Some will reject this argument precisely because they accept something like functionalism for the other physical determinables, and hence deny the thought experiment about the non-electromagnetic force--they will say that if the laws are structurally the same, the properties are literally the same.

I think there is a way to counter the above argument by pointing out a disanalogy between location and other fundamental physical determinables (this disanalogy goes against the spirit of this post, alas). Let's say we live in an Einsteinian world. A Newtonian world still might have been actual. But, plausibly, the Newtonian world's "mass" is a different determinable from our world's mass. Here's why. In our world, mass is the very same determinable as energy (one could deny this by making it a nomic coextensiveness, but I like the way of identity here). In the Newtonian world "mass" is a different determinable from "energy". Therefore either (a) Newtonian "mass" is a different determinable from mass, or (b) Newtonian "energy" is a different determinable from energy, or (c) both (a) and (b). Of these, the symmetry of (c) is pleasing. More generally, it is very plausible that fundamental physical determinables like mass-energy, charge, charm or wavefunction are all law bound: you change the relevant laws (namely, those that make reference to these determinables) significantly, and you don't have instances of these determinables.

But location does not appear to be law bound. "Location" in a Newtonian spacetime and a relativistic spacetime are used univocally. You can have a set of really weird laws, with a really weird 2.478-dimensional space (for fractional dimensions, see, e.g., here), and yet still have location. Maybe there are some formal constraints on the laws needed for locations to be instantiated, but intuitively these are lax.

Plausibly, natural (in the David Lewis sense of not being gerrymandered) physical determinables that are not law bound are functional. If location is a natural physical determinable, which is very plausible on an absolutist view of spacetime, then it is, plausibly, functional. I think an analogous argument can be run on relationism, except that the fundamentality constraint is a bit less plausible there.

One might question the claim that natural physical determinables that are not law bound are functional. After all, if the claim is plausible with the "physical", isn't it equally plausible without "physical"? But the dualist denies the claim that natural determinables that are not law bound are functional. For instance, awareness seems to be a natural determinable (whose determinates are of a form like being aware of/that ..., and nothing else), but the dualist is apt to deny that it's functional.

In any case, one interesting result transpires from the above. It is an important question whether location is law bound. If we could resolve that, we would be some ways towards a good account of spacetime (if it is law bound, proposals like this one might have some hope, if based on a better physics). The account I give above of law boundedness is rather provisory, and a better account is also needed.

Evolution and the Principle of Sufficient Reason

One of my graduate students recently suggested in discussion that if one rejects the Principle of Sufficient Reason (PSR), our knowledge of evolution may be undercut. We can use this insight to generate an ad hominem argument for the PSR. Most atheists and agnostics, and some theists, believe that there is a naturalistic evolutionary explanation of the development of the human species from a single celled organism. I claim that they are not justified in believing this unless they accept the PSR. But I also think (though I shan't argue for it here) that if the PSR is true, then the cosmological argument works, and God exists.

For consider what the argument for thinking that there is such an evolutionary explanation could be. We might first try an inductive argument. Some features of some organisms can be given naturalistic evolutionary explanations. Therefore, all features of all organisms can be given naturalistic evolutionary explanations. But this argument is as bad as inductive arguments come. The error in the argument is that we are reasoning from a biased sample, namely those features for which we already have found an explanation. Such features are only a small portion of the features of organisms in nature—as always in science, what we do not know far exceeds what we know.

Once we admit the selection bias, the argument becomes: “All the features of organisms for which we know the explanation can be explained through naturalistic evolutionary means.” There are at least two things wrong with this argument. The first is that it might just be that naturalistic explanations are easier to find, and hence it is no surprise that we first found those explanations that are naturalistic. But even if one could get around this objection, it would not obviate the need for the PSR. For the argument at most gives us reason to accept the claim that those features that have explanations have naturalistic evolutionary explanations. The inductive data is that all the explanations of biological features that we have found are naturalistic and evolutionary. The only conclusion that can be drawn without the PSR is that all the explanations of biological features that there are are naturalistic and evolutionary, not that all biological features have naturalistic evolutionary explanations.

A different approach would be to suppose that natural occurrences have naturalistic explanations, and evolution is the only naturalistic form of explanation of biological features that we know of, so that it is likely that the development of the human race has a naturalistic evolutionary explanation. But what plausibility is there in the claim that natural occurrences have naturalistic explanations if one does not accept the PSR for contingent propositions? After all, if it is possible for contingent propositions to simply fail to have an explanation, what reason do we have for confidence that at least those contingent propositions that report natural occurrences have explanations? If “natural occurrence” is taken as entailing the existence of a naturalistic explanation, the argument for an evolutionary explanation of the development of the human race becomes question-begging in its assumption that the development was a natural occurrence. But if “natural occurrence” is taken more weakly as a physical event or process, whether or not it has a natural explanation, then the naturalness of the occurrence does not give us reason to think that the occurrence has an explanation, much less a naturalistic one, absent the PSR. If we had the PSR in play, we could at least tryt o use some highly defeasible principle about the cause being ontologically like the effect, so that if the effect is natural, the cause is likely such as well.

Consider a final way to justify the evolutionary claim. We have good inductive reason to think that everything physical obeys the laws of physics. But everything that is governed by the laws of physics has a naturalistic explanation. Hence, the development of the human race has a naturalistic explanation, and an evolutionary one is the best candidate we have.

The claim that everything that obeys the laws of physics has a naturalistic explanation, however, has not been justified. The claim was more plausible back when we thought that everything could be explained in a Newtonian manner, but even then the claim could be falsified. Consider John Norton’s ball-on-dome example. We have a rigid dome, on the exact top of which there sits a perfectly round ball, and the dome is in a constant downward gravitational field of acceleration g. The dome’s is rotationally symmetric, and its height as a function of the distance r from its central axis is h=(2/3g)r^3/2. It turns out to be consistent with Newtonian physics that the ball should either remain still at the top of the dome or start to roll down in any direction whatsoever, in the absence of any external forces. One might wonder how this squares with Newton’s second law—how there could be an acceleration without an external force. It turns out, however, that because of the shape of the dome, in the first instant of the ball’s movement, its acceleration would be zero, and after that it would have an acceleration given by the gravitational force. The physics would fail to explain the ball’s standing still at the top of the dome or the ball’s moving in one direction or another; it would fail to explain this either deterministically or stochastically. Thus, even Newtonian physics is not sufficient to yield the claim that everything that obeys the laws of physics can be explained in terms of the laws of physics.

And I doubt we do any better with non-Newtonian physics. After all, we do not actually right now know what the correct physics is going to be, and in particular we do not know whether the correct physics will make true the claim that everything that obeys the laws of physics can be explained in terms of the laws of physics. Besides, surely it would be an implausible claim that justification for the claim that the human race developed through evolutionary means depends on speculation about what the final physics will be like.

I do not have an argument that there is no other way of arguing for the evolutionary claim absent the PSR. But, intuitively, if one weren’t confident of something very much like the PSR, it would be hard to be justifiedly confident that no features of the human species arose for no reason at all—say, an ape walked into a swamp, and out walked a human, with no explanation of why.