# Reductionism vs Galilean invariance

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

1. The only thing that really exists are particles whose only properties are position, velocity, mass, and maybe some other quantities like that.
2. 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.

## 11 thoughts on “Reductionism vs Galilean invariance”

1. This is very important and very clearly expressed. Thank you.

Actually, I’d go even further than you do. Neither X nor V is meaningful in a two-particle universe. You call X the “relative distance between the particles” — but relative to what? Since that distance is the only spatial quantity in the two-particle universe, there is no way to measure it or to say in any non-arbitrary way whether it is increasing or decreasing. Without X, there can obviously be no V — but even if we somehow could determine that X were increasing or decreasing, there could still be no meaningful V, since temporal quantities, no less than spatial ones, can be quantified only by comparison to something else. If only one change is taking place, the speed of that change cannot be defined.

A minimum of three particles is necessary in order for position and velocity to exist.

2. angliablog |

Dear Bonald,

I think it could be argued that you have the usual map-terrain problem that is somehow always ignored by scholastics. You seem think that statements like “movement is a property of a particle” are to be taken as truths as real as the world, that words and statements like this can be taken as 100% total absolute real truths or falseshoods. But any scientist schooled in operationalism or general semantics would say you are putting the map before the terrain and take statements too literally, too real. They would say that – at least in the operationalist logic accepted by many scientists – a statement merely has “truth content” and its “truth content” is simply its ability to predict observations, and only observations should be seen ultimately real, statements are just models which are to some extent useful for prediction and to some extent not, and not some kind of ultimate truths or falseshoods. Their view is basically less rationalist than yours.

I do not necessarily agree with this because there are some problems here, like this view seems to be too dependent on the correspondence theory of truth and after Quine I tend to agree more with the coherence theory of truth.

But anyway it is IMHO a valid argument that you scholastics take statements as something too real, and binary true or false, and do not notice map-terrain problems, In other words you have a too rationalist view of the universe: thinking it is intelligible and thus there must be a particular combination of words that describes everything perfectly. This is a valid concern and should be concerned. I think your rationalist epistemology could use some justification.

http://plato.stanford.edu/entries/operationalism/

http://en.wikipedia.org/wiki/General_semantics

• If no statement is entirely true then ‘no statement is entirely true’ is not entirely true, in which case there must be some statement or other that is entirely true, so that ‘no statement is entirely true’ is entirely false.

The imprecision in thought, expression and modeling that General Semantics rightly notices is due, not to the impossibility of knowledge, but to poorly made thoughts, expressions and models. If we are careful to define terms appropriately, specify domains properly, and qualify statements adequately, and so forth, we can say and know truths.

• To your first point: why confine ourselves to the two unrealistic extremes of formal logic, “all” and “none”? I’d be happy sticking to something like “the vast majority of” or “practically none”.

What do you mean by knowledge? If you mean knowledge expressed in words – as opposed to knowledge as experience of a thing etc. – you are basically assuming the most important thing you are trying to prove: a theistic universe, for only a theistic universe can be rational on the level of words. I mean, truth is, ultimately, just the way things are outside in the world, not the words in our minds and mouths. Basically if your point is that there are certain words that are as real as the things out there, and not just a vague map or model of them, if words can have the the same level of reality as reality itself, then it could only happen in a universe with a designer who also, kind of, thinks in words. But this something you are trying to prove with all your posts, so it is not really good to just assume it.

Parallel: you are trying to buy a house and go visit and and make a sketch of the layout, a map, a blueprint for yourself. This plan will always be less accurate and less real than the real house. But if you can find the original blueprints the designer designed and the builders followed, those will be as real or even more real than the house itself, because the house is a copy of the original blueprint, while your plan is a copy of the house. Thus, if you have this kind of epistemological optimism, rationalistic optimism, about the accurady of _verbal_ Reason, you are assuming what you are trying to prove, namely that there is a Blueprint and there is a Designer and when we think we are basically trying to figure out, as Einstein said, God’s thoughts. But you are trying to prove this, so it should not be assumed. I know I am asking something close to impossible – pretty much all our philosophy and even science was throughout most of its history built on this very same assumption that there is some original blueprint and our quest for knowledge is looking for it and not just drawing one on our own! So I understand if it is hard.

(BTW I am the same commenter as above, I am just re-setting a WordPress account not used since about 2006, sorry.)

• Shenpen, if ‘no statement is entirely true’ is false, then some statements are entirely true. Most are not, as you say, but some definitely are. Consider, for example, the statement, “‘No statement is entirely true’ is false.” I have just demonstrated that it is simply true, because its contrary is self-refuting.

This demonstration leaves you all the room you want for distinguishing between true and false statements, for a healthy skepticism about statements, for distinguishing between map and territory, etc.

But note then that “some statements are false” is trivial.

As to distinguishing between the truth of statements and the reality of things in the world, well, granted; but, again, this is trivially true. It does not follow from its truth either that we cannot apprehend the reality of things in the world or that we cannot make true statements. If we are careful in respect to apprehension and ratiocination, we can know reality; and if we are careful in respect to terms and logic, we can make true statements (such as, e.g., the statement, “some truths may be stated”).

None of this has anything directly to do with the question whether theism is true. I happen to agree with you that there could be no objective reality, no cosmos, if there were no God; so that, if there were no God, and ergo no cosmos, neither therefore could there be such things as us. Yet here we are.

But that’s a different argument.

3. Bill |

Feser’s article, linked in Feser’s blog post, is pretty interesting. My reaction to the “problem of motion” when I first understood it was to think that 1) Aristotle had just gotten the physics wrong, so that 2) we could just fix it by identifying what Aristotle calls motion with what we call acceleration. And this seems like a way to incorporate many potential physics as well. Identify stasis with “what happens when nobody is doin’ nuttin’ ” to the particle and motion with changes in state. Metaphysics is supposed to be unfalsifiable, so this seems like an unproblematic plan to me, at least on that score.

This is one of the possibilities Feser considers in his article, but he mentions in passing that Garrigou-Lagrange considers and rejects this possibility. Does anyone know why G-L rejects this move?

4. 691 |

I have no stake in the reductionism argument, but you do make a mistake along the way.

” 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.”

I disagree with this statement. it is not compatible with the current mathematical understanding and interpretation of space. We can think unambiguously and rigorously about specific points, independent of any labeling or choice of inertial reference.

We can say that the particle is moving, because for any particular frame of reference, for any given labeling you choose, it will be seen to be moving. While it may be impossible to avoid an arbitrary choice along the way, we can still think of intrinsic properties as those properties that will appear no matter which choice we have made. In one choice, the particle appears to move from A to B; in another it appears to move from C to D, in another it appears to move from Y to Z.

There are durable, robust mathematical phenoma that require arbitrary choices to state, explain or interpret but nonetheless are independent (up to some appropriate equivalence) of that arbitrary choice.

What you seem to be getting at towards the end is that Galilean invariance (or any other type of symmetry) does not give us complete freedom to choose coordinates as we see fit. There is some other underlying structure that must be preserved. There are constraints as to how we can label our points or choose our frame of reference. Symmetry gives us an equivalence relation between two configurations but not all configurations are equivalent up to the given symmetry.

• Bonald |

There is one frame, as good as any other, where the particle appears to “move” from A to A. Whether a particle is or is not moving is a frame-dependent statement, and the only way to nonarbitrarily answer the question is to refer to some other structure that singles out a particular frame, i.e. to break the symmetry. This can’t come from the laws of physics themselves, since they have the symmetry in question, so it must come from relation to other objects–the center of mass frame of the local system, the frame in which the CMB is isotropic, etc.

• 691 |

I see, we’re arguing about space vs space-time. But your reductionism argument falls apart if we consider particles to be worldlines in space-time. Spacetime is made up of identifiable points. Worldlines do have intrinsic properties. Up to reparametrization, we can uniquely identify the points it occupies in space-time. We can, say, define a cotangent vector to the worldline at every point, from which we can compute the momentum in any inertial frame of reference by an appropriate linear projection.

• Bonald |

Hi 691,

I’m starting to think that we’re talking past each other a bit–at least one of us not seeing the point that the other is trying to make. Sure, it’s true that we can identify the events the particle passes through (even at a single spatial slice, we can identify whether an observer is exactly at the particle’s position or not, so if this is a problem, it’s already there in 3D), and we even have a natural parameter (the proper time, up to an offset) to label them with, and a particle has a distinct 4-velocity and 4-momentum, but whether or not the particle is at rest (“staying at the same spatial point at different times”) is still a frame-dependent statement. We need a second timelike vector to say that. In its absence, we can just say that the worldline is timelike, which is a frame-invariant statement. If you think that I keep switching back and forth between a 3D and a 4D picture, you are right, but that seems inescapable where philosophical discussions of motion are concerned, and one of my points was to highlight the ambiguities involved. Thank you for your input, by the way. I did come up with my argument fairly quickly, and it’s possible that I’ve missed some important points.

• Bonald |

Hi 691,

On second thought, I think I do see what you’re getting at. I’ve been making a big deal about the inability to unambiguously identify spatial points at different times, and you’re saying that if I take a consistently 4D view, the issue doesn’t even arise–one simply *doesn’t* identify different spacetime events. This is a good point, a very good point. It shows that my argument really is attached to theories with absolute time (and I’m sorry for confusing matters by inconsistently throwing out relativistic issues in the comments), which may be okay for interpreting Newtonian physics (my original goal), but it certainly limits its overall interest. In a spacetime picture, I could still argue that a single worldline really only makes time meaningful, and the 4-velocity is nothing but the generator for advancing the internal clock on the particle, but I certainly haven’t presented an argument (even to myself) that this is the preferred interpretation.

Thank you very much for your time.

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