[discoinverno]

I don't understand your point, I never said that "everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it", I just wanted to mention that "continuity of spacetime is a convenient approximation" may be a correct sentence in the context of quantum gravity.

Also, for what is worth, in QM the space of wavefunctions can also be finite dimensional (for instance the Hilbert space of a spin 1/2 particle).

[mjburgess]

minimum lengths arent relevant to whether things are continuous or not. these arent related.

[lupire]

That's literally the definition of continuity.

You have an object at position p, and the behaviors of the system are discretely different between P and P + h, without an intermediary at P+h/2.

[mjburgess]

And that's not what a "minimum length" in this case means. We're not talking about space having a minimum unit. We're talking, at best, about (presumably massive) objects having a minimum extension in space .

Even with a "minimum length" (in this specific sense), you have an object at position p, and can (move/observe) it at any p+dx continuously.

importantly, the question is whether the best theories of physics in a world with a minimum extension-in-space require continuous mathematics, and there's nothing about this plank length to suggest they wouldnt

[simiones]

Discreteness would mean that there exists some base distance p such that the distance between any two objects is Np, with N being a natural number (and any surface is some Mp^2 and any volume is Qp^3 and so on). Continuity is simply the opposite of that. It could be that objects can be at arbitrary real-valued distance d from each other, but that d > p is a precondition for any other law of physics.

By contrast, discretness has various unintuitive mathematical properties that mean it's not easy to fit into some other theories (particularly those relying on differential equations).