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Comment by mjburgess | Hacker News Reader

You're talking about minimum lengths, not discrete spacetime.

It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous.

There being a minimum length doesnt mean 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.

Whenever people try to do philosophy of physics the inevitable place everyone lands at is a series of false equivocations, often caused by the language of physics being ambiguous and polysemous. But "minimum length" here does not mean a sort of grid length.

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I agree with all your claims here in the literal sense, but suppose there's a minimum length, then it would seem to be at least theoretically possible to use discrete mathematics to formulate an approximation to the "real number formulation of physics"?

The fact that we don't have already a full system using discrete maths doesn't mean it is impossible, because our current system is based on a long tradition of belief in real numbers, and assuming physical space is continuous.

I'd argue (admittedly unhelpfully) that unless we have actually tried to formulate physics using discrete mathematics and found a barrier that we prove unequivocally that it is impossible to overcome, we can't claim that physics must be formulated using real numbers/continuous math. There's a difference between "we don't know how to do this" vs "we know we can't do this".

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).

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

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.

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

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).

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