>>12816573Physical space? I’m not confident enough one way or another to tell you, though it’s a point of practicality that physicists don’t consider anything less than the Planck length.
If we’re talking about the value of geometry on a smooth vs discrete level, there’s a fuckton of geometry that is invariant whether or not the spaces are smooth or not, so it really comes down to expressing fundamental geometric ideas correctly.
>>12816614Basically. See above. Continuous theory is so good that we even “port it” over to the language of discrete spaces when we’re dealing with graphics, algorithms, engineering, etc.. The cool thing is that we can go in the other direction too. This is all predicated on the idea that we don’t study discrete geometry as approximations in the same way as numerical analysis approximates real numbers, but that discrete geometry complements the smooth setting as long as you preserve the key properties. Here’s a cool survey on the matter:
https://www.ams.org/publications/journals/notices/201710/rnoti-p1153.pdfWhat this suggests to me is that if we write down the ideas carefully and precisely enough, it won’t matter that much if space is discrete or continuous.
>t. inb4 CS midwifI studied both physics and CS, and given differential geometry’s key role in mainstream physics, it’s a really big deal if discrete and continuous phenomena are strongly connected by both a canonical mathematical atlas (kek) and in a canonically physical way. Can’t say I understand the latter though
