Reading Edward Feser’s article on inertia and how it should be reconciled with Aristotelian natural philosophy got me thinking. What does Newtonian kinematics really mean? Let’s pretend that Newtonian physics is the exact truth and ask what that would imply about the nature of reality. As we know, the rise of quantitative sciences was accompanied by the rise of a reductionistic atomism which the new science was thought to imply. Even today, it is thought a “scientific” way of thinking to suppose
- The only thing that really exists are particles whose only properties are position, velocity, mass, and maybe some other quantities like that.
- Composite bodies are just combinations of these particles, and the particles themselves have some sort of ontological priority over their arrangement. They are “more real” than the forms of composite bodies, which are just consequences of the particles’ positions.
Feser and others have done good work arguing that neither Newtonian physics nor any other conceivable empirical scientific theory prove the above postulates. I think we can go farther and say that they are actually incompatible with Newtonian physics.
First thing: what specifies the state of a particle in Newtonian kinematics? It is the particle’s position x and velocity v, or equivalently the position x and conjugate momentum p. The laws of motion are either 2 first-order differential equations for x and p or one second-order ODE for x. Either way, the velocity/momentum is a part of the particle’s state in a way the acceleration–which is fixed by outside forces–isn’t. Thus, the “problem of motion” can’t be “Why does the particle move instead of staying in place in the same state?” since if the particle were to instantly stop, that would be as much a change in state as if it were to continue at constant velocity. In one case, p and v change, in the other case x changes, and these would both be “motion” in the Aristotelian sense.
But is a particle moving with constant velocity changing its state? Suppose the universe were completely empty except for one particle moving at constant speed. Does it make sense to say that this particle is moving? Not if we take Galilean invariance seriously. It makes no sense to say that really the particle is moving from point A to point B to point C etc. because the labelling of points is arbitrary. Space is not made of identifiable points the way a floor is made of identifiable tiles. Saying that something is staying at the same point in space is not like saying that it is remaining in contact with some physical object; the identification of two events at different times as having the same spatial location is just a statement about the coordinate system you are using, not something about physical reality. Thus, a particle alone in its own universe cannot move.
Now, anyone is free to say that there could be a preferred inertial frame in reality that just doesn’t show up in the laws of physics. This would be importing structure into the interpretation of the theory that does not naturally connect to the theory at all. We can do this if some other metaphysical commitment forces us to, but I like to think of adding structure to accommodate my preconceptions as an act of last resort. Let us go ahead and take Galilean invariance as reflecting the actual nature of physical reality, as most physicists and non-scholastic philosophers would encourage us to do. I will argue that the result is actually agreeable to us.
What would it mean to incorporate Galilean invariance at the ontological level. It would mean that, since there is no unique way of identifying a particle’s position and momentum, these are not actually properties of the particle at all. They are arbitrary but useful mathematical devices, like the choice of zero in energy or the choice of gauge in electromagnetism.
Now, suppose there are two particles in the universe moving at constant velocity. Now it is possible to unambiguously identify motion–the particles can be getting closer or farther away. Does this mean that, although a particle in isolation has no position or momentum, it acquires these as soon as it shares the universe with something else, even if that something doesn’t interact with it? That would be absurd; the presence of a noninteracting neighbour can’t change what the particles’ intrinsic properties are. What’s more, x, v, and p are still meaningless. What is meaningful is the relative distance between the particles X and how fast this is changing, V = dX/dt.
So now we have a universe with some properties, X and V, that can change. To what do these properties belong? They are not derivative of properties of the individual particles or of their separate relations to space. True, for a given choice of coordinates we can compute X = x1 – x2, but that choice is arbitrary. The position and momentum are purely relational properties, and if we are to ascribe them as properties to any one entity, it must be to the 2-particle system as a whole.
A final question about the proper interpretation of Newtonian physics is whether we should regard space as a real entity or just a sum of the relations between the particles. I lean toward the former, for the following reason. Suppose there were only one spatial dimension, so all particles must lie on a single line. (This assumption won’t affect the conclusions; it just makes the argument easier to understand.) Suppose there were three particles, and we label their positions as x1, x2, and x3. Labels, as I’ve been saying, are arbitrary, and one could just as well have chosen x1+c, x2+c, and x3+c for any c. The real things are the relations X12 = x1-x2, X13=x1-x3, X23=x2-x3. Notice that these relations aren’t independent. We must have X13=X12+X23. Why? If space were just a shorthand for a collection of relations, I don’t see why X12, X13, and X23 couldn’t be anything. Apparently, something is constraining them, and we say that this comes from the system being embedded in “space”. The key property about space in this theory is how it constrains geometric relations. Notice that we must say that the system itself is embedded, not the individual particles. If it were the particles that had separate relations with space, this would be equivalent to giving the particles unambiguous positions (which, after all, are just relations to space), and this would violate the spirit of Galilean invariance. So, again, the most basic aspect of particles, their existing and moving in space, is an inherently collective phenomenon.
In these ways, we see that the combined reductionist proposition, that composites are ontologically posterior to elementary particles and their associated (x,p), is incompatible with Newtonian (and all later) physics. Form, at least the elementary form of the arrangement of the system, is equally primitive.