The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics.

The use of mathematical functions (continuous) to describe the evolution over time of quantities such as position, velocity, acceleration, energy, has been introduced since Newton's time. However, when using a calculator, the mathematical functions of physical use are subjected to a necessary discretization which involves a certain error. My question is: is the reality in which the mentioned physical quantities are discrete? Could we conceive the environment in which a body moves, with a certain trajectory, like a three-dimensional screen composed of Pixels? In this case, the use of the integral calculation would result in a mathematical error, in exactly the opposite way to the discretization process that is conducted in a computer.

My physics professor said that reality is continuous, but I do not think that this concept can be assimilated by the human mind. I do not want to come to the treatment of space-time, but I believe that the paradoxes of Zeno are sufficient to agree that the physical greatness with which we deal every day is of a discrete nature.

Quantum mechanics confirms that entities such as energy and speed should be understood as discrete (just think of the "quantum" of energy), therefore it is possible that my question can be answered already in this. However, since the school years the use of continuous mathematics is taught but not justified. Is it possible that the universe is discrete, but composed of such a high number of stencils that any error is insignificant for classical mechanics, which deals with the macroscopic world?

  • $\begingroup$ It is highly possible, although once you make the universe discrete, classical mechanics won't be able to give any sufficiently correct result. Loop quantum gravity is proceeding in this direction and string theory also is trying to convince us that particles are discrete vibrating strings. Till now, experiments have not shown any deviation from a continuous universe. So your question of a classically discrete universe makes little sense. :") $\endgroup$ Mar 6 '18 at 17:03
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    $\begingroup$ The justification for assuming a continuous spacetime is that it fits all the observations and it is simpler than the alternative. $\endgroup$
    – Javier
    Mar 6 '18 at 18:19

Both classical and quantum physics use mathematical models that assume space and time are continuous. Within those models, Zeno's paradoxes do not represent obstacles; assuming continuous space and time, calculus resolves the "paradoxes" with no trouble. And indeed, we can get from point $A$ to point $B$, so maybe a "correct" theory should be able to resolve such "paradoxes".

But classical and quantum physics are just models. They are only useful insofar as they explain our observations of the world around us. Your physics professor evidently overstated things: we only know that "reality" is continuous if our current models accurately reflect reality — which is not something we can say for sure. In fact, we know that classical and quantum physics are incomplete at some point.

If your model does a better job of explaining some phenomenon, then it may indeed be a better model. It may even be "true" — whatever that means. So yes, it's certainly conceivable that "reality" is pixellated. But you'll have to do more work to convince people that that's a useful way of thinking about things.

  • $\begingroup$ Actually, the electric field is also schematised as a continuum, but instead is composed of photons. Yet calculations performed by the integral method are considered accurate. Perhaps because the mistake committed is definitely negligible? If you work with a gravity discrete field, the trajectory of a satellite will be different from the one calculated with a continuous model. Maybe that, anyway, the discrete "resolution" of the universe is so high that we can't perceive the difference between a continuous result and a discrete one? $\endgroup$ Mar 6 '18 at 20:14
  • $\begingroup$ About Zenos paradox, the solution based on the converging numerical series is founded on the continuous, and is necessary to introduce the Limit concept to close the historical argue. For this reason i think that there is not an easy solution that human mind can understand without maths. But maths is not the real world. About the solution offered by quantum mechanics, i like it. And infact the quantum theory talk about discretization. $\endgroup$ Mar 6 '18 at 20:25
  • $\begingroup$ Modern quantum theory says that photons are excitations of the quantum electrodynamical field, which — in this model — exists in the continuum of spacetime. So even though the excitations may be discrete in certain ways, there's nothing discrete about spacetime — in this model. Again, feel free to create your own model, and come up with concrete testable predictions. That's just how science works. But so far, science says that our models involving continuous space and time are doing pretty well. $\endgroup$
    – Mike
    Mar 6 '18 at 20:36
  • $\begingroup$ "But maths is not the real world." That's probably true, but we use maths to model the real world as we experience it. These are just models. No good scientist will claim that they're "right" in any fundamental sense — just that they seem to work pretty well in a lot of ways; they give us concrete predictions, and so far they've passed a lot of the tests of those predictions. $\endgroup$
    – Mike
    Mar 6 '18 at 20:39

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