3 ms·

There's uncountably many more noncomputables than computables too, but the noncomputables hardly crop up, even though they come along with the reals also.

semi-extrinsic 7y ago
Well, you might reasonably expect that a theory describing how to compute numbers will give you more computable numbers than uncomputable ones, no?
- jerf 7y ago
  It's worse than you make it sound, though: https://www.youtube.com/watch?v=5TkIe60y2GI https://www.youtube.com/watch?v=5TkIe60y2GI In short, it isn't just that we have more computables than non-computables; there are categories of numbers like "normal" that nominally contain "all the numbers" but we can barely even name a handful.
  - wiz21c 7y ago
    really interesting video.
    - AceyMan 7y ago
      This YT link is to a recent (2019) video on the Numberphile channel with Matt Parker as presenter, just so people know what level of quality to expect (14m26s runtime). And, yeah, that's one of my favorites in the STEM genre. I can't wait until my son (now nine) is ready to watch along with me.
- roywiggins 7y ago
  We can't approximate noncomputables at all. If there was an effective algorithm for doing so, they would be computable. The real numbers are a bit spooky.
  - inflatableDodo 7y ago
    Also, if you add any noncomputable to a computable, the result is noncomputable. Basically, since the reals were created by René Descartes, in a misunderstood attempt at numerical sarcasm, the number line has become infested with conceptual prions, or 'priorns' as the tabloids call them nowadays, that are worryingly both everywhere and conversely, far too small to detect. If these priorns get into the brain tissue of a working mathematician, the neural substrates responsible for higher order mathematical abstractions and preferences for knitwear, can start to come up with really stupid number definitions, such as binary reals where every even nth digit is the nth digit of pi and every odd nth digit is from the result of an idealised coin flip. If this should unfortunately occur to a mathematician in your area, the kindest thing to do is to try and find them a role in quantative trading and tell them that they are going to be working on something really important. The banking sector subsidises daycare office environments providing these fictional job roles, as a social service for those infected by priorns, as a way of providing thanks to the wider mathematical community for inventing all the numbers that banking relies on. It is unfortunate to have to resort to such deceit, however by that stage in the pathology, mathematicians become constantly irrational and fintek is therefore provided as a safe and enclosed habitat where they cannot damage anything particularly critical to wider society. They may even thank you for the opportunity, the poor deluded things.
jerf 7y ago
If you did want to salvage a "calculus without real numbers" the computables are probably more promising. All of the criticisms of the article still entirely apply to them, but there's a better chance you could come up with substitute definitions that may yield some useful concepts; they're closed on a more useful operations. I believe this is an actual studied area of math. But it's certainly more complicated than using real numbers. We may not teach, ahem, "real" real numbers to high school students, but you'll get to it fairly early in dedicated college math courses at the undergrad level.
- roywiggins 7y ago
  There's some work on it apparently. https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
- inflatableDodo 7y ago
  I've wondered before if there could be a well defined set of numbers between the integers and reals (perhaps with its own class of infinite set) that nicely excludes the uncomputables. If there is, I suspect that it might be what the universe is actually using and if I was going to go looking for it, I'd follow the breadcrumbs from umbral moonshine and go sniffing around the monster group.
  - roywiggins 7y ago
    Other than the rationals? Those are all computable. Or the algebraics. Whether there's an infinity between the naturals and the reals depends on the continuum hypothesis, which is independent of the usual axioms of set theory, so you can pick.

semi-extrinsic 7y ago

Well, you might reasonably expect that a theory describing how to compute numbers will give you more computable numbers than uncomputable ones, no?

jerf 7y ago
It's worse than you make it sound, though: https://www.youtube.com/watch?v=5TkIe60y2GI https://www.youtube.com/watch?v=5TkIe60y2GI In short, it isn't just that we have more computables than non-computables; there are categories of numbers like "normal" that nominally contain "all the numbers" but we can barely even name a handful.
- wiz21c 7y ago
  really interesting video.
  - AceyMan 7y ago
    This YT link is to a recent (2019) video on the Numberphile channel with Matt Parker as presenter, just so people know what level of quality to expect (14m26s runtime). And, yeah, that's one of my favorites in the STEM genre. I can't wait until my son (now nine) is ready to watch along with me.
roywiggins 7y ago
We can't approximate noncomputables at all. If there was an effective algorithm for doing so, they would be computable. The real numbers are a bit spooky.
- inflatableDodo 7y ago
  Also, if you add any noncomputable to a computable, the result is noncomputable. Basically, since the reals were created by René Descartes, in a misunderstood attempt at numerical sarcasm, the number line has become infested with conceptual prions, or 'priorns' as the tabloids call them nowadays, that are worryingly both everywhere and conversely, far too small to detect. If these priorns get into the brain tissue of a working mathematician, the neural substrates responsible for higher order mathematical abstractions and preferences for knitwear, can start to come up with really stupid number definitions, such as binary reals where every even nth digit is the nth digit of pi and every odd nth digit is from the result of an idealised coin flip. If this should unfortunately occur to a mathematician in your area, the kindest thing to do is to try and find them a role in quantative trading and tell them that they are going to be working on something really important. The banking sector subsidises daycare office environments providing these fictional job roles, as a social service for those infected by priorns, as a way of providing thanks to the wider mathematical community for inventing all the numbers that banking relies on. It is unfortunate to have to resort to such deceit, however by that stage in the pathology, mathematicians become constantly irrational and fintek is therefore provided as a safe and enclosed habitat where they cannot damage anything particularly critical to wider society. They may even thank you for the opportunity, the poor deluded things.

jerf 7y ago

If you did want to salvage a "calculus without real numbers" the computables are probably more promising. All of the criticisms of the article still entirely apply to them, but there's a better chance you could come up with substitute definitions that may yield some useful concepts; they're closed on a more useful operations. I believe this is an actual studied area of math. But it's certainly more complicated than using real numbers. We may not teach, ahem, "real" real numbers to high school students, but you'll get to it fairly early in dedicated college math courses at the undergrad level.

roywiggins 7y ago
There's some work on it apparently. https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
inflatableDodo 7y ago
I've wondered before if there could be a well defined set of numbers between the integers and reals (perhaps with its own class of infinite set) that nicely excludes the uncomputables. If there is, I suspect that it might be what the universe is actually using and if I was going to go looking for it, I'd follow the breadcrumbs from umbral moonshine and go sniffing around the monster group.
- roywiggins 7y ago
  Other than the rationals? Those are all computable. Or the algebraics. Whether there's an infinity between the naturals and the reals depends on the continuum hypothesis, which is independent of the usual axioms of set theory, so you can pick.