Fun problem I was thinking through last night: how do you abnitialize the concept of pi if you're in a "snap to grid" universe that has only rational increments of the base quantnization.
@munin Why not? You can still go to the limit of large sizes. Similarly in our universe we are nearly always considering setups where the Planck length (or even diameter of a molecule) is infinitesimally small.
nb. I said "snap to grid" - e.g. all materials would have some 'elasticity' in them, so to speak, that would not necessarily appear to be fully consistent in-universe - instead you'd have a discontinuous "this length of fence can enclose this much area" table without any clear ability to infer intermediate values.
I'm trying to see if I can break math a lil bit with the right circumstances :3
> instead you'd have a discontinuous "this length of fence can enclose this much area" table without any clear ability to infer intermediate values.
But we do live in such a world too! A fence in our world has some number of molecules in its circumference, their connections can just stretch somewhat (and usually there's a mixture of connections of very different lengths, and the fence is nonnegligibly wide, etc.). The only reason we don't think in those terms is that our fences are much larger than that ~quantum.
If you posit that the quantum is macroscopic in the proposed world from the POV of some intelligent entities, then I doubt whether they would have enough volume to store enough information in their minds, unless you posit some very-high-information-content indivisible (i.e. of the size of one fundamental cube) thing.
It's a universe I'm building to break math with, so I'm not concerned about the biological details yet lol
Hm~ I then don't understand what level of self-consistency you want (nit: it's not a biological detail, because anything that processes information will have such problems). Another thing to think of in such a world is the question of straight lines: are all directions permissible in contexts where something spreads along a line? What's the distance in e.g. electrostatic attraction, or what are the directions a light ray can travel in (I vaguely remember that restricted wave propagation directions is something we do encounter in our world in some crystal structures btw.)?
If you haven't already seen anything by Greg Egan, you might enjoy his novels and stories. Many of them posit worlds with different physics (and two even with, after a fashion, different mathematics) or different ~biology (with the source of the difference unspecified) and try to extrapolate into them. Depending on what is the difference in a given story, the extrapolation is either very detailed (e.g. Clockwork Rocket), very vague (e.g. Teranesia), or somewhere inbetween (e.g. Morphotropic).
Actually, @gregeganSF in case he finds this topic interesting. tl;dr how could a world look like if "legal positions" were quantized to a grid at a scale that's macroscopic (or nearly macroscopic) to its inhabitants.
Oh yeah, I've read a bit of Eagan; he's got some fascinating ideas.
@robryk
you could theoretically approximate it, sure, but a 'snap to grid' universe would not have a 'natural' concept of continuous functions