3 ms·

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

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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo ago
        Pi, i and e show up with apparent perfect "precision" in all kinds of physics. Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve. Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory. I think "precision" is the wrong way to look at what can mean something or not. I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers. From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.) What is true about all these numbers is not precision, but that they emerge from a finite number of relationships. They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.) And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values. So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail. Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.
- Nevermark 3mo 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.

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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo 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 3mo ago
        Pi, i and e show up with apparent perfect "precision" in all kinds of physics. Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve. Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory. I think "precision" is the wrong way to look at what can mean something or not. I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers. From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.) What is true about all these numbers is not precision, but that they emerge from a finite number of relationships. They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.) And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values. So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail. Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.
Nevermark 3mo 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.