The world is full of correlations: situations where if we know one thing, we can infer another. In classical physics these correlations are local, which means they are comprehensible. In quantum physics they are non-local, which means they are incomprehensible.
To understand the incomprehensibility it's worth understanding the comprehensible case first, which is what I'm going to be focused on here. For all that it's comprehensible, it's still hard to understand. Everything I say below is completely counter-intuitive. Any feeling of understanding you get from the description is the result of one person--Isaac Newton--making the mysterious, the bizarre, the non-obvious comprehensible to the likes of you and me.
Nothing I'm saying here was understood by anyone before the mid-1600s: Socrates, Plato, Aristotle, Origen, Augustine, Aquinas, Luther... nobody. None of the "great thinkers" of antiquity grasped any of this. There might have been other things they failed to grasp as well.
Consider two pucks on an air-hockey table that are of different sizes, one stationary, one moving toward it.
"Stationary" and "moving" are relative terms: to someone walking by they are both moving. To someone moving with precisely the same velocity as the "moving" one it is stationary and the "stationary" one is on a reciprocal path toward it.
This fact can be used to help us analyze the collision that is about to happen.
We'll look at the simplest case, which is when the line of motion passes through the centre of both pucks, as shown in the usefully labelled diagram below:
There are a bunch of incredibly counter-intuitive things about this picture. The first is that the pucks have labels "M" and "m", which stand for their mass, which Newton defined as "the quantity of matter and the measure of the same, arising from the density and volume conjointly." (I'm quoting from an English translation of Newton's Principia from memory, so may not be entirely accurate, but the sense is right.)
We take this for granted today, but up until the mid-1600s no one had any idea what the appropriate measure of the "quantity of stuff" was. There were people who thought it was the volume, and others the density. Very few thought it was the weight, which is at least related to the mass by the force of gravity.
Newton was able to conceptualize an entirely new quantity that was different from all of these things--mass--and realized its most important role was resistance to change. More massive objects are harder to get moving. Less massive objects are easier to get moving.
But what is this thing, "moving"?
The problem that Newton faced when wondering "Why do things move?" is he had to invent an entire set of concepts, each of which was in part defined by the others. There was nothing linear or reductionist about it, which is why it took such an extraordinary mind to come up with it. Reductionism is the only mechanism we have to bring the world within the scope of the human mind: holistic approaches are anti-human, and violate the most fundamental fact about us, which is we can only keep three to five things in mind at once. Newton had to focus obsessively for years to be able to hold everything he needed in his mind to separate motion and its causes into three things: the measure of things, the measure of motion, and the measure of the thing that changes motions.
These are: mass, momentum, and force.
Mass, as I've said, is the measure of "stuff". Momentum is one measure of motion. For Newton it was literally "the quantity of motion", just like a volume is the quantity of a liquid, say. Weirdly, in our common systems of units we have no unit for momentum, which is equal to mass times velocity.
For Newton, though, mass times velocity was the quantity of motion, and the thing that changed that quantity was force: objects at rest tend to remain at rest, and objects in motion tend to remain in motion, unless they are acted upon by a force. Nor does force come out of nowhere: Newton's third law says "for every action there is an equal and opposite reaction". That implies that if one mass changes its state of motion by gaining or losing momentum, there must be another mass somewhere that does the same in the opposite. Momentum is conserved.
There is a second conserved quantity in (some) collisions: energy. Collisions that conserve energy are said to be "elastic" because the objects involved spring back into shape after they run into each other.
Energy was a latecomer to the physics party, but eventually displaced momentum as what we think of as a "fundamental" quantity. It's worth remembering, though, that "fundamental" means "fundamental to our understanding or description of the world". It's no more fundamental to the world itself than the colour blue is. The world is the world: it is itself, independently of us, in its entirety. We analyze it and break it down and describe it in ways we can comprehend, and that's good, but our descriptions are not the world, and there is no reason to believe the world has any particular inherent hierarchical structure of being. It just is. All of it. Concepts like energy describe its being to us, that's all. The Tao that can be spoken…
Conservation of energy was discovered via what is now a famous toy: Newton's Cradle. This is a device with half a dozen heavy balls hanging from wires so they touch. If you pull one back and let it go, the one on the far end bounces away while the ones in the middle remain motionless, even though they are free to move. Conservation of momentum alone leaves the future of the system undetermined: one ball at the end could jump with the same velocity as the ball that struck the far side, or two balls could jump with half that velocity, or three with a third, and so on.
Kinetic energy--the energy of motion--is determined by the square of the velocity, and equal to (m/2)v**2. There is only one motion that conserves both energy and momentum: the one where a single ball takes on the same velocity as the impactor (assuming all the balls have equal mass.)
We can use conservation of energy and momentum to predict what happens when the little puck hits the big puck. The velocity of the big puck after the collision is V and the little puck is v', and conserving energy and momentum gives us:
(m/2)v**2 = (M/2)V**2 + (m/2)v'**2
mv = MV + mv'
We now have two equations in two unknowns (V and v') and while it takes a little work to solve them:
v′ = v(m − M)/(m + M)
V = 2mv/(m+M)
The thing about this is: the two velocities after the collision are perfectly correlated: if we know one, we know the other (and we know the initial velocity of the incoming little puck). If we know v’ we can solved the first equation above for v and use that to compute V: knowledge of one is knowledge of the other.
All it takes is one number to let us infer something about the others no matter how long after the collision or how far apart the pucks are. There is no limit to this inference, because absent external forces the two pucks will go sailing off into infinity, assuming our air hockey table is sufficiently large. This is no big deal in classical physics, because it is simply a consequence of the fact that the collision has a particular, definite, single, outcome. There is a fact of the matter as to the result of the interaction of the two pucks.
But this has profound consequences when we get to the quantum description of reality, which allows more than one classical outcome to a simple collision like this, but which still retains the correlations implied by conservation of energy and momentum in every case.
That fact is what ultimately allows us to infer that there is not just more to the universe than we see: there is more to the universe than we can see.
This was very clear to me. Thinking about the last post and your explanation of causation, I am starting to understand The idea that you're getting at. I especially your point that the two velocities after the collision are perfectly correlated. Of course I know this, and I know a lot about statistics. Most of the statistics I use are based in algebra: I used Analysis of Variance in my PhD work. But I'm starting to see where you're going with this. Look forward to more posts about this.