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It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed number

It is too bad we don't have a pithy familiar term for the computable / definable / "nameable" / constructible reals. The most general class of undisputed numbers, consistent with the forms we actually use to represent quantities and perform numerical operations. I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.

Dylan16807 2mo ago
> I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers. They're worse. Just having another dimension is significantly more relevant to reality. And the reals also ruin the word "normal".
- tialaramex 2mo ago
  I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.
  - Dylan16807 2mo ago
    As far as we can tell, there is nothing resembling a number with infinite digits in the real world. It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.
    - tialaramex 2mo ago
      Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and across mathematics we find this ratio again, and again, and again, it's something fundamental but it clearly isn't rational. Likewise for the square root of 2 and for Euler's Number. These numbers are ever so real and yet they sure fucking look normal to me. If you assure me they are not normal, but you can't prove it, I shall not believe you.
      - Dylan16807 2mo ago
        The real world doesn't have any of those numbers, only approximate matches. My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.
        tialaramex 2mo ago
        > You can do math by hand with more precision than actually exists in the real world. This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.
        singularity2001 2mo ago
        what he probably meant is that if you write down pi with 200 digits there's nowhere in the universe where you can find pi with that precision
        tialaramex 2mo ago
        Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC. But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.
        Dylan16807 2mo ago
        The most precise situations I can think of involve impossibly perfect measurements of volume. And even there, okay a cubic meter is 10^105 cubic planck units and the visible universe is 10^186. Finite math can easily throw a million digits at any problem. How do you reach a point where you need reals to describe actual things?
    - Nevermark 2mo ago
      I agree but with a slightly different definition. There are no numbers in the real world, that require and infinitely long definition. 1/3 has infinitely many digits in decimal, but has a finite definition. So its good. I would write down the opposite, a definition of an uncomputable, unnameable real, but there are not enough atoms in the universe (multi-verse, ...) to being to do that.
Diogenesian 2mo ago
I don't think it makes sense to say anything except "computable real" - the computable / uncomputable distinction seems totally immaterial for the purposes of most real analysis, or even "pathological" topology and set theory involving R (except for puzzles directly involving computability). And the "interesting" transcendental computable reals are a bit of a grabbag. The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem.
- Nevermark 2mo ago
  In this case, the term "uncomputable" also means "undefinable with less than infinite symbols". As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable". Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist. 1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits. Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.) (Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)

Dylan16807 2mo ago

> I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers. They're worse. Just having another dimension is significantly more relevant to reality. And the reals also ruin the word "normal".

tialaramex 2mo ago
I actually think both "real" and "normal" are helpful in building the correct intuition about how complicated the world we inhabit is rather than how simple we want it to be.
- Dylan16807 2mo ago
  As far as we can tell, there is nothing resembling a number with infinite digits in the real world. It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.
  - tialaramex 2mo ago
    Of course "having infinite digits" isn't from the real world because it's about an artifice, our choosing to represent numbers with digits. But the distinction between the Rationals and the Reals isn't about those digits. The discovery that there's some fixed ratio between the diameter of a circle and its circumference is fascinating and yet though we can't (AFAIK) prove it's normal that ratio sure looks normal and across mathematics we find this ratio again, and again, and again, it's something fundamental but it clearly isn't rational. Likewise for the square root of 2 and for Euler's Number. These numbers are ever so real and yet they sure fucking look normal to me. If you assure me they are not normal, but you can't prove it, I shall not believe you.
    - Dylan16807 2mo ago
      The real world doesn't have any of those numbers, only approximate matches. My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.
      - tialaramex 2mo ago
        > You can do math by hand with more precision than actually exists in the real world. This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.
        singularity2001 2mo ago
        what he probably meant is that if you write down pi with 200 digits there's nowhere in the universe where you can find pi with that precision
        tialaramex 2mo ago
        Are you sure? Naively sure, you won't find sufficiently enormous circles and even if you could such a huge circle won't have the Pi ratio here, that's a property of Euclidean space and we don't live in a Euclidean space, ours is slightly off IIRC. But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.
        Dylan16807 2mo ago
        The most precise situations I can think of involve impossibly perfect measurements of volume. And even there, okay a cubic meter is 10^105 cubic planck units and the visible universe is 10^186. Finite math can easily throw a million digits at any problem. How do you reach a point where you need reals to describe actual things?
  - Nevermark 2mo ago
    I agree but with a slightly different definition. There are no numbers in the real world, that require and infinitely long definition. 1/3 has infinitely many digits in decimal, but has a finite definition. So its good. I would write down the opposite, a definition of an uncomputable, unnameable real, but there are not enough atoms in the universe (multi-verse, ...) to being to do that.

Diogenesian 2mo ago

I don't think it makes sense to say anything except "computable real" - the computable / uncomputable distinction seems totally immaterial for the purposes of most real analysis, or even "pathological" topology and set theory involving R (except for puzzles directly involving computability). And the "interesting" transcendental computable reals are a bit of a grabbag. The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem.

Nevermark 2mo ago
In this case, the term "uncomputable" also means "undefinable with less than infinite symbols". As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable". Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist. 1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits. Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.) (Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)