4 ms·
It's not individual numbers that matter, but the properties of sets of them and functions over them. You can define a Reimann integral over computable numbers w
by dooglius 5y ago
It's not individual numbers that matter, but the properties of sets of them and functions over them. You can define a Reimann integral over computable numbers without changing much, but not the more general Lebesgue integral. This means you won't get e.g. the nice properties of L2 spaces which are used heavily in quantum mechanics.
- xscott 5y agoI'm not sure if you'll see this reply since it's been several days, but I appreciate the answer. I majored in math, but I never stumbled on the Lebesgue integral, and I don't know enough quantum to understand the subtlety there either. I'll have to take your word for it that L2 spaces don't work with just computable numbers, but I can't see why yet. All of that to say, I'm just admitting my ignorance and not arguing at all. Still, I think it's interesting that if Reals are truly needed to model the universe (physics), then there must be some case where computables are insufficient. And if the universe isn't computable, that says something amusing and deep about determinism and whether we can ever succeed in modelling it. :-)
- dooglius 5y agoOne of the major pillars of measure theory is countable subadditivity, which serves to allow one to work with countably infinite series in a nice way. This is pretty much unworkable in a countable space like the computables. That isn't to say we really "need" the reals, merely that math hasn't come up with a fully-featured alternative framework that physics can use. I know there are people working on "computable measure theory" in recent years, and there's no reason why the universe should use a mathematical framework that is easy for we humans to construct.