5 ms·

Yep - This really is just a random number problem.. it is only interesting in the sense that Pi is something that has been captivating in public math perception

Yep - This really is just a random number problem.. it is only interesting in the sense that Pi is something that has been captivating in public math perception. Any random number would have the same characteristics.. It is however something fun to do in a homelab outside of the normal learning and playing around, and that has merit for me. Hobbies are hobbies in part because of the interest, joy and appreciation they drive.

Spivak 2y ago
In one sense you're right. The set of normal numbers has measure 1 so if you choose a real number randomly it will be normal and have the same properties. In another sense you're very wrong. The set of uncomputable numbers also has measure 1. So while there are lots of normal numbers Pi belongs to a pretty exclusive club of numbers we can actually look at. It's not a particularly deep result but the fact that the set of computable numbers is countable will never not make me existential. Every number that we can actually express, that we will ever know, is no bigger than the whole numbers.
- drexlspivey 2y ago
  How do you generate a random irrational number?
  - chowells 2y ago
    Generate a random Cauchy sequence. One trivial way to do this is to output a random string of digits. It's not perfect, but it works. Approximately 0% of random Cauchy sequences will be computable, so long as your random number generator doesn't have self-correlations.
    - danbruc 2y ago
      Would this not require generating an infinite number of random numbers which you can not do? Or can you actually generate a random Cauchy sequence from a finite set of random numbers?
      - Spivak 2y ago
        I think the answer is yes and no. Let's say we knew we had a normal number, say we could prove pi is normal. Then the set of sequences defined by the decimal expansion of pi starting at the nth place for all n in the natural numbers would be uniformly random over all decimal expansions. I'm going this route because it in theory allows us to define each sequence by virtue of a single number rather than an infinite sequence of randint(0, 9). This I think would be our best case if it were possible. But you would have to select a number uniformly in the range [0, inf) which... you can't.
        danbruc 2y ago
        We actual know some normal numbers [1] but they seem to all be synthetic. I would guess they have a lot of local correlation because of the way they are constructed and the comment I responded to explicitly mentioned no self-correlations - not sure what exactly that means - but it probably indicates that being a normal number is not a strong enough condition for using its digits as a sequence of random digits. [1] https://en.wikipedia.org/wiki/Normal_number#Properties_and_examples https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...
      - chowells 2y ago
        "which you can not do" is an interesting assertion. In some sense it's trivially true, of course. Generating infinite random numbers would take infinite time. In fact, I'm quite happy to say that approximately 0% of real numbers can exist, for that reason. But in contrast to that, I'm also happy to grant the existence of constructions with countably infinite length when discussing theories outside of computability. So in that context, is it possible to generate an infinite string of random numbers? Well, it is so long as you believe you can generate any amount random numbers at all. Just repeat the process, whatever it is. Is that a reasonable thing to believe? That's probably a philosophical question at this point. I'm certainly not equipped to answer it. But assuming it is possible allows for some interesting math, which I support for its own sake. This is all sort of reminiscent of a mathematician friend's stance on the axiom of choice. If you were to tell them a vehicle is only guaranteed to not explode by the axiom of choice, they wouldn't use it. On the other hand, assuming the axiom of choice is true leads to some useful math, even if it's sort of sketchy ontologically. It's a lot like programming. Sometimes you can't fix the bugs in the underlying system and you just code around them to get your stuff working.

Spivak 2y ago

In one sense you're right. The set of normal numbers has measure 1 so if you choose a real number randomly it will be normal and have the same properties. In another sense you're very wrong. The set of uncomputable numbers also has measure 1. So while there are lots of normal numbers Pi belongs to a pretty exclusive club of numbers we can actually look at. It's not a particularly deep result but the fact that the set of computable numbers is countable will never not make me existential. Every number that we can actually express, that we will ever know, is no bigger than the whole numbers.

drexlspivey 2y ago
How do you generate a random irrational number?
- chowells 2y ago
  Generate a random Cauchy sequence. One trivial way to do this is to output a random string of digits. It's not perfect, but it works. Approximately 0% of random Cauchy sequences will be computable, so long as your random number generator doesn't have self-correlations.
  - danbruc 2y ago
    Would this not require generating an infinite number of random numbers which you can not do? Or can you actually generate a random Cauchy sequence from a finite set of random numbers?
    - Spivak 2y ago
      I think the answer is yes and no. Let's say we knew we had a normal number, say we could prove pi is normal. Then the set of sequences defined by the decimal expansion of pi starting at the nth place for all n in the natural numbers would be uniformly random over all decimal expansions. I'm going this route because it in theory allows us to define each sequence by virtue of a single number rather than an infinite sequence of randint(0, 9). This I think would be our best case if it were possible. But you would have to select a number uniformly in the range [0, inf) which... you can't.
      - danbruc 2y ago
        We actual know some normal numbers [1] but they seem to all be synthetic. I would guess they have a lot of local correlation because of the way they are constructed and the comment I responded to explicitly mentioned no self-correlations - not sure what exactly that means - but it probably indicates that being a normal number is not a strong enough condition for using its digits as a sequence of random digits. [1] https://en.wikipedia.org/wiki/Normal_number#Properties_and_examples https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...
    - chowells 2y ago
      "which you can not do" is an interesting assertion. In some sense it's trivially true, of course. Generating infinite random numbers would take infinite time. In fact, I'm quite happy to say that approximately 0% of real numbers can exist, for that reason. But in contrast to that, I'm also happy to grant the existence of constructions with countably infinite length when discussing theories outside of computability. So in that context, is it possible to generate an infinite string of random numbers? Well, it is so long as you believe you can generate any amount random numbers at all. Just repeat the process, whatever it is. Is that a reasonable thing to believe? That's probably a philosophical question at this point. I'm certainly not equipped to answer it. But assuming it is possible allows for some interesting math, which I support for its own sake. This is all sort of reminiscent of a mathematician friend's stance on the axiom of choice. If you were to tell them a vehicle is only guaranteed to not explode by the axiom of choice, they wouldn't use it. On the other hand, assuming the axiom of choice is true leads to some useful math, even if it's sort of sketchy ontologically. It's a lot like programming. Sometimes you can't fix the bugs in the underlying system and you just code around them to get your stuff working.