Many posters to this Thread do not seem to understand that the classical reality of our senses is built on quantum entities/processes which are not continuous.
If you mean particles, that part is well understood generally don't you think?
At the quantum level probabilistic laws apply. The probabilistic laws & the Uncertainty Principle indicate that particle motion cannot be modeled using continuous paths.
The statistics themselves relate a continuum of probability densities. The particle can always be anywhere within a small continuum. Of course it's hard to trace trajectories for particles jumping in an out of existence. Yet even these show conservation of momentum, a property generally consigned to the continuum of spacetime, at least as far as kinematics is concerned.
Various Posts use the following analogy as an argument for a continuous reality.
if we investigated this movie closer, we would notice that digital creates gaps in reality that we can prove were not in reality.
Actually, the analogy is an argument for a discreet reality.
A movie is nothing more than a simulation of real scenes.
There is no evidence supporting the notion of continuity at the quantum level.
You mean, like wave-particle duality? Or do you mean the uncertainty that a particle can be anywhere within the spacetime continuum plus or minus a very small but continuous range of positions within the band of uncertainty?
Since the macro level is built on quantum entities & processes, the appearance of continuity is an illusion.
You mean world-scale reality is an illusion because it's just the sum of its parts, that is, Planck length sized domains where particles appear out of thin air, in ambiguous states and virtual in nature, a world of weirdness and paradoxes?
Just as the discrete frames of film or video recordings produce the illusion of continuous action to our eyes, the discrete processes at the quantum level produce an illusion of continuous action at the classical level.
But reality is nothing like a movie, and a movie is at best only a simulation of something real (or fictional).
We view planetary orbits as continuous because our mind considers a planet to be a solid object
Except of course for the gas planets, or satellite photos of earth with its clouds, oceans and terrains.
which can be modeled by the continuous motion of its center of mass.
But reality is not a model. So each particle interacts with every other particle according to real laws that consign their motion to a real trajectory that is not only continuous but relativistic.
Actually, a planet is mostly empty space.
Quantum or continuum?
There is no good reason to believe that there is a center of mass which moves along a continuous path.
Which is why you offered it as a mere model.
At the quantum level particles are moving randomly in that mostly empty space.
And yet they are orbiting the sun predictably.
Calculation of the path of the center of mass would require integrating position/mass variables using a continuous time variable.
Time and space, you mean. Same for the particles that are orbiting the sun. But you don't have to integrate. You can just sum all the infintesimals - I mean quanta.
This would require that the position/motion of each individual quantum particle be describable by continuous functions.
Fortunately such a calculation has no practical significance as it would be quite laborious.
Such functions do not exist: The path of quantum level particles cannot be described by continuous functions.
There are at least two components that determine the particle's position. The first is the result of accounting for its angular acceleration due to gravity during orbit around the sun. That reality manifests as a continuum. The second component is the result of trying to resolve the particle's position so finely, on the order of a Planck length, that the measurement becomes meaningless in relation to the scale of an astronomical unit (times 2π).
BTW, I think the idea of center of mass works well for very large numbers of particles, since the sum of all uncertain quantities x ± Δx over a very large population averages out the uncertainty . . . most of the time.