4 ms·

I should've been more clear. The Reals are very unsettling because there are supposed to be so many of them, but nobody has used, named, or described more than

by xscott 2y ago

I should've been more clear. The Reals are very unsettling because there are supposed to be so many of them, but nobody has used, named, or described more than a few which aren't computable. They are angels dancing on a pin which we've never seen and probably don't need for any of applied mathematics.

Tainnor 2y ago
> and probably don't need for any of applied mathematics Basically everyone who uses mathematics in some applied capacity happily uses the real numbers.
- xscott 2y ago
  No, all the numbers most anyone actually uses are computable, and it'd say something weird about determinism if they weren't. The word "need" was carrying some weight, as in necessary vs sufficient.
  - Tainnor 2y ago
    It's not about individual numbers. Most applied maths fields (including physics) use the real numbers and the properties of real numbers as a complete field are implicitly assumed all the time (e.g. for differential equations). It's only ever some mathematicians, computer scientists and philosophers who care about the fact that most of them aren't computable. If you want to argue that one could derive a theory of physics etc. that doesn't rely on the real numbers that's one thing. That's probably possible although I doubt that it would be very elegant. But it's simply not what people use.
    - xscott 2y ago
      > If you want to argue that one could derive a theory of physics etc. that doesn't rely on the real numbers that's one thing. That wasn't my original intent, but I think it's a legitimate stance. The Reals are bizarre (Banak Tarski, etc...), and we practically never use the non-computable ones. > It's only ever some mathematicians, computer scientists and philosophers who care about the fact that most of them aren't computable. Lol, who else WOULD care? :-) > That's probably possible although I doubt that it would be very elegant. That's an aesthetic judgment. Someone else might argue that sticking to the numbers we actually use and can calculate is simpler/cleaner/more elegant/whatever. And besides, I'm guessing you could replace ℝ with T (for Turing) and most of the proofs would go just fine. (Although the Computables have plenty of weirdness to them too...)
      - Tainnor 2y ago
        > And besides, I'm guessing you could replace ℝ with T (for Turing) and most of the proofs would go just fine. This is wrong. LUB doesn't hold for the computable reals, and this means that a lot of classical proofs straight up don't work anymore. That's not to say you can't build a different kind of real analysis built only on the computable reals, some mathematicians have tried to do similar things in the past, but it's going to be a different framework with not exactly the same theorems and not exactly the same proofs; for example, there are no discontinuous functions anymore. There's some more discussion here: https://www.reddit.com/r/math/s/LSL487ncag https://www.reddit.com/r/math/s/LSL487ncag
        xscott 2y ago
        Yeah, I'm sure there are some holes (poor attempt at a pun), but you already know you can get arbitrarily close to whatever bound with rationals, much less computables, so it seems like someone better educated than me could shore that up. Anyways, the computables have unfortunate properties too (not knowing if you'll eventually need to carry for addition, or when to stop testing for equality), so I'm left wondering if some genius in the future will come up with a new way to build the numbers which are more satisfying. Thank you for the Reddit link.

by xscott 2y ago

Tainnor 2y ago

> and probably don't need for any of applied mathematics Basically everyone who uses mathematics in some applied capacity happily uses the real numbers.

xscott 2y ago
No, all the numbers most anyone actually uses are computable, and it'd say something weird about determinism if they weren't. The word "need" was carrying some weight, as in necessary vs sufficient.
- Tainnor 2y ago
  It's not about individual numbers. Most applied maths fields (including physics) use the real numbers and the properties of real numbers as a complete field are implicitly assumed all the time (e.g. for differential equations). It's only ever some mathematicians, computer scientists and philosophers who care about the fact that most of them aren't computable. If you want to argue that one could derive a theory of physics etc. that doesn't rely on the real numbers that's one thing. That's probably possible although I doubt that it would be very elegant. But it's simply not what people use.
  - xscott 2y ago
    > If you want to argue that one could derive a theory of physics etc. that doesn't rely on the real numbers that's one thing. That wasn't my original intent, but I think it's a legitimate stance. The Reals are bizarre (Banak Tarski, etc...), and we practically never use the non-computable ones. > It's only ever some mathematicians, computer scientists and philosophers who care about the fact that most of them aren't computable. Lol, who else WOULD care? :-) > That's probably possible although I doubt that it would be very elegant. That's an aesthetic judgment. Someone else might argue that sticking to the numbers we actually use and can calculate is simpler/cleaner/more elegant/whatever. And besides, I'm guessing you could replace ℝ with T (for Turing) and most of the proofs would go just fine. (Although the Computables have plenty of weirdness to them too...)
    - Tainnor 2y ago
      > And besides, I'm guessing you could replace ℝ with T (for Turing) and most of the proofs would go just fine. This is wrong. LUB doesn't hold for the computable reals, and this means that a lot of classical proofs straight up don't work anymore. That's not to say you can't build a different kind of real analysis built only on the computable reals, some mathematicians have tried to do similar things in the past, but it's going to be a different framework with not exactly the same theorems and not exactly the same proofs; for example, there are no discontinuous functions anymore. There's some more discussion here: https://www.reddit.com/r/math/s/LSL487ncag https://www.reddit.com/r/math/s/LSL487ncag
      - xscott 2y ago
        Yeah, I'm sure there are some holes (poor attempt at a pun), but you already know you can get arbitrarily close to whatever bound with rationals, much less computables, so it seems like someone better educated than me could shore that up. Anyways, the computables have unfortunate properties too (not knowing if you'll eventually need to carry for addition, or when to stop testing for equality), so I'm left wondering if some genius in the future will come up with a new way to build the numbers which are more satisfying. Thank you for the Reddit link.